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tabs="inherit"><f family="Arial" charset="0"><u><inlineAttr line-through="false"><c val="#808000"><link href="./basics.xmcd" title="Links to Section Contents" popup="false"><sp count="5" />UTILITIES</link></c></inlineAttr></u></f></p></text></region><region region-id="114" left="36" top="72.75" width="105" height="13.5" align-x="36" align-y="84" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Sous-titre" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Maxima and Minima</p></text><indexes><phrase>maxima and minima</phrase><phrase>utilities, maxima and minima</phrase><phrase>minima and maxima</phrase></indexes></region><region region-id="613" left="66" top="103.5" width="87.75" height="12.75" align-x="82.5" align-y="114" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">localmax(x, y, [w]) </p></text><indexes><phrase>localmax</phrase><phrase>functions, localmax</phrase></indexes></region><region region-id="1144" left="180" top="103.5" width="102.75" height="12.75" align-x="196.5" align-y="114" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">localmax(x, y, M, [w]) </p></text><indexes><phrase>localmin</phrase><phrase>functions, localmin</phrase></indexes></region><region region-id="1142" left="66" top="121.5" width="86.25" height="12.75" align-x="82.5" align-y="132" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">localmin(x, y, [w]) </p></text><indexes><phrase>localmax</phrase><phrase>functions, localmax</phrase></indexes></region><region region-id="1145" left="180" top="121.5" width="101.25" height="12.75" align-x="196.5" align-y="132" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="functions" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">localmin(x, y, M, [w]) </p></text><indexes><phrase>localmin</phrase><phrase>functions, localmin</phrase></indexes></region><region region-id="676" left="36" top="158.25" width="293.25" height="96" align-x="36" align-y="168" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The routines <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">localmax</c></inlineAttr></b></f> and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">l</c></inlineAttr></b></f><f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">ocalmin</c></inlineAttr></b></f> compare adjacent values in a data set to determine local maxima and minima. The optional window value, w, determines how many points on each side of the point in question must also be smaller or larger for the point to be considered maximum or minimum. Larger windows will help give true maxima for noisy data sets. These functions work for both 2D and 3D data sets.</p></text></region><region region-id="569" left="36" top="271.5" width="54.75" height="12.75" align-x="36" align-y="282" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Local peaks</p></text><indexes><phrase>local peaks</phrase><phrase>peaks, local</phrase></indexes></region><region region-id="571" left="36" top="290.25" width="287.25" height="48" align-x="36" align-y="300" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">It's sometimes useful to know where the local maxima and minima are in a data set before doing subsequent processing. 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item-idref="12" /></region><region region-id="734" left="300" top="537" width="30" height="12.75" align-x="312.75" align-y="546" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>w</ml:id></ml:eval></math><rendering item-idref="13" /></region><region region-id="1066" left="222" top="560.25" width="145.5" height="36" align-x="222" align-y="570" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Set the window width to the place where only real maxima and minima are found.</p></text></region><region region-id="1215" left="72" top="603" width="120.75" height="12.75" align-x="108.75" align-y="612" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Maxima</ml:id><ml:apply><ml:id>localmax</ml:id><ml:id>Data1</ml:id><ml:id>w</ml:id></ml:apply></ml:define></math><rendering item-idref="14" /></region><region region-id="1216" left="222" top="603" width="119.25" height="12.75" align-x="258" align-y="612" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Minima</ml:id><ml:apply><ml:id>localmin</ml:id><ml:id>Data1</ml:id><ml:id>w</ml:id></ml:apply></ml:define></math><rendering item-idref="15" /></region><region region-id="701" left="24" top="630" width="344.25" height="192.75" align-x="24" align-y="630" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="16" /><rendering item-idref="17" /></region><region region-id="610" left="36" top="824.25" width="313.5" height="48" align-x="36" align-y="834" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">The graph shows the local maxima and minima using the actual data values. If you wish to know the absolute maximum for approximate interim values, take the numerical derivative of the <b><u><link href="./nonparam.xmcd" title="Links to section" popup="false">interpolated curve</link></u></b> and find the places where it is zero.</p></text></region><region region-id="1240" left="36" top="895.5" width="171.75" height="12.75" align-x="36" align-y="906" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Finding functional peaks and troughs</p></text><indexes><phrase>finding functional peaks and troughs</phrase><phrase>functional peaks and troughs, finding</phrase><phrase>troughs</phrase><phrase>peaks, functional</phrase></indexes></region><region region-id="704" left="36" top="914.25" width="330" height="36" align-x="43.5" align-y="924" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you know a functional form for your data, perhaps by using <b><u><link href="./regression.xmcd" title="Links to section" popup="false">regression</link></u></b> techniques, you can find maxima and minima by searching for the roots of the first derivative:</p></text></region><region region-id="1058" left="66" top="962.25" width="186.75" height="27.75" align-x="86.25" align-y="978" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:function><ml:id>f</ml:id><ml:boundVars><ml:id>x</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:real>1</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>3</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>10</ml:real></ml:apply><ml:apply><ml:pow /><ml:parens><ml:apply><ml:plus /><ml:id>x</ml:id><ml:real>1</ml:real></ml:apply></ml:parens><ml:real>4</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="18" /></region><region region-id="1059" left="36" top="998.25" width="302.25" height="36" align-x="36" align-y="1008" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">There are numerous ways to do this in Mathcad. The method shown below is applicable to polynomials. First, find the polynomial coefficients using live symbolics:</p></text></region><region region-id="745" left="66" top="1046.25" width="145.5" height="62.25" align-x="93" align-y="1080" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>coeff</ml:id><ml:symEval><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:id>f</ml:id><ml:id>x</ml:id></ml:apply></ml:lambda></ml:apply><ml:command><ml:sequence><ml:id>coeffs</ml:id><ml:id>x</ml:id></ml:sequence></ml:command><ml:command><ml:sequence><ml:id>float</ml:id><ml:real>2</ml:real></ml:sequence></ml:command><ml:symResult><ml:matrix rows="4" cols="1"><ml:real>1.4</ml:real><ml:real>-.80</ml:real><ml:real>-1.8</ml:real><ml:real>.40</ml:real></ml:matrix></ml:symResult></ml:symEval></ml:define></math><rendering item-idref="19" /></region><region region-id="1147" left="0" top="1116" width="6000" height="6" align-x="0" align-y="1116" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1060" left="36" top="1130.25" width="309" height="24" align-x="36" align-y="1140" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Use the vector of coefficients as an argument to the polyroots function:</p></text></region><region region-id="756" left="66" top="1179" width="102" height="12.75" align-x="93" align-y="1188" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>zeros</ml:id><ml:apply><ml:Polyroots method="companion-matrix" /><ml:id>coeff</ml:id></ml:apply></ml:define></math><rendering item-idref="20" /></region><region region-id="757" left="222" top="1162.5" width="73.5" height="45.75" align-x="248.25" align-y="1188" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>zeros</ml:id></ml:eval></math><rendering item-idref="21" /></region><region region-id="752" left="66" top="1203" width="47.25" height="18" align-x="80.25" align-y="1212" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>x0</ml:id><ml:apply><ml:indexer /><ml:id>zeros</ml:id><ml:real>0</ml:real></ml:apply></ml:define></math><rendering item-idref="22" /></region><region region-id="1062" left="144" top="1203" width="47.25" height="18" align-x="158.25" align-y="1212" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>x1</ml:id><ml:apply><ml:indexer /><ml:id>zeros</ml:id><ml:real>1</ml:real></ml:apply></ml:define></math><rendering item-idref="23" /></region><region region-id="758" left="48" top="1236" width="254.25" height="219.75" align-x="48" align-y="1236" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><plot disable-calc="false" item-idref="24" /><rendering item-idref="25" /></region><region region-id="1065" left="36" top="1466.25" width="318.75" height="60" align-x="36" align-y="1476" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Bump points on the graph of the function f(x) are zero crossings for the derivative, which are found by polyroots. You could use the root function for any functional form to find roots one at a time, or you can try the following program, based on the symbolic processor's root finder:</p></text></region><region region-id="1244" left="66" top="1547.25" width="180" height="104.25" align-x="96" align-y="1602" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>Roots</ml:id><ml:symEval><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:id>f</ml:id><ml:id>x</ml:id></ml:apply></ml:lambda></ml:apply><ml:command><ml:sequence><ml:id>solve</ml:id><ml:id>x</ml:id></ml:sequence></ml:command><ml:symResult><ml:matrix rows="3" cols="1"><ml:real>-1</ml:real><ml:apply><ml:plus /><ml:apply><ml:div /><ml:real>11</ml:real><ml:real>4</ml:real></ml:apply><ml:apply><ml:mult style="narrow-dot" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>4</ml:real></ml:apply><ml:apply><ml:pow /><ml:real>65</ml:real><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply><ml:apply><ml:minus /><ml:apply><ml:div /><ml:real>11</ml:real><ml:real>4</ml:real></ml:apply><ml:apply><ml:mult style="narrow-dot" /><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>4</ml:real></ml:apply><ml:apply><ml:pow /><ml:real>65</ml:real><ml:apply><ml:div /><ml:real>1</ml:real><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:apply></ml:matrix></ml:symResult></ml:symEval></ml:define></math><rendering item-idref="26" /></region><region region-id="1243" left="0" top="1668" width="6000" height="6" align-x="0" align-y="1668" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="764" left="66" top="1683" width="276" height="201" align-x="125.25" align-y="1692" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>CriticalPoints</ml:id><ml:program><ml:for><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:apply><ml:id>last</ml:id><ml:id>Roots</ml:id></ml:apply></ml:range><ml:program><ml:localDefine><ml:apply><ml:indexer /><ml:id>out</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>0</ml:real></ml:sequence></ml:apply><ml:apply><ml:indexer /><ml:id>Roots</ml:id><ml:id>i</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:apply><ml:indexer /><ml:id>out</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>1</ml:real></ml:sequence></ml:apply><ml:apply><ml:id>f</ml:id><ml:apply><ml:indexer /><ml:id>Roots</ml:id><ml:id>i</ml:id></ml:apply></ml:apply></ml:localDefine><ml:localDefine><ml:id>x</ml:id><ml:apply><ml:indexer /><ml:id>Roots</ml:id><ml:id>i</ml:id></ml:apply></ml:localDefine><ml:localDefine><ml:id>MinOrMax</ml:id><ml:apply><ml:id>signum</ml:id><ml:apply><ml:derivative /><ml:lambda><ml:boundVars><ml:id>x</ml:id></ml:boundVars><ml:apply><ml:id>f</ml:id><ml:id>x</ml:id></ml:apply></ml:lambda><ml:degree><ml:real>2</ml:real></ml:degree></ml:apply></ml:apply></ml:localDefine><ml:ifThen><ml:apply><ml:equal /><ml:id>MinOrMax</ml:id><ml:real>-1</ml:real></ml:apply><ml:localDefine><ml:apply><ml:indexer /><ml:id>out</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>2</ml:real></ml:sequence></ml:apply><ml:str>Maximum</ml:str></ml:localDefine></ml:ifThen><ml:ifThen><ml:apply><ml:equal /><ml:id>MinOrMax</ml:id><ml:real>1</ml:real></ml:apply><ml:localDefine><ml:apply><ml:indexer /><ml:id>out</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>2</ml:real></ml:sequence></ml:apply><ml:str>Minimum</ml:str></ml:localDefine></ml:ifThen><ml:otherwise><ml:localDefine><ml:apply><ml:indexer /><ml:id>out</ml:id><ml:sequence><ml:id>i</ml:id><ml:real>2</ml:real></ml:sequence></ml:apply><ml:str>Inflection</ml:str></ml:localDefine></ml:otherwise></ml:program></ml:for><ml:id>out</ml:id></ml:program></ml:define></math><rendering item-idref="27" /></region><region region-id="765" left="66" top="1900.5" width="198" height="45.75" align-x="124.5" align-y="1926" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>CriticalPoints</ml:id></ml:eval></math><rendering item-idref="28" /></region><region region-id="1071" left="282" top="1900.5" width="73.5" height="45.75" align-x="308.25" align-y="1926" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id>zeros</ml:id></ml:eval></math><rendering item-idref="29" /></region><region region-id="725" left="36" top="1963.5" width="233.25" height="12.75" align-x="36" align-y="1974" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="subsubtitle" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Finding local and global maxima and minima in 3D</p></text><indexes><phrase>finding local and global maxima and minima in 3D</phrase><phrase>maxima, finding local and global</phrase><phrase>minima, finding local and global</phrase><phrase>3D, finding local and global maxima and minima in</phrase></indexes></region><region region-id="1121" left="36" top="1988.25" width="324" height="36" align-x="36" align-y="1998" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Using a demonstration function with one local maximum and 4 local minima, we can use the same empirical <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">localmax</c></inlineAttr></b></f> and <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">localmin</c></inlineAttr></b></f> functions for 3D matrices:</p></text></region><region region-id="787" left="66" top="2030.25" width="157.5" height="27.75" align-x="98.25" align-y="2046" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:function><ml:id>g</ml:id><ml:boundVars><ml:id>x</ml:id><ml:id>y</ml:id></ml:boundVars></ml:function><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:apply><ml:minus /><ml:apply><ml:plus /><ml:apply><ml:neg /><ml:apply><ml:pow /><ml:id>x</ml:id><ml:real>2</ml:real></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>3</ml:real><ml:id>x</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:real>2</ml:real><ml:apply><ml:pow /><ml:id>y</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="default" /><ml:apply><ml:div /><ml:real>3</ml:real><ml:real>4</ml:real></ml:apply><ml:id>y</ml:id></ml:apply></ml:apply><ml:real>100</ml:real></ml:apply></ml:define></math><rendering item-idref="30" /></region><region region-id="1127" left="66" top="2067" width="37.5" height="12.75" align-x="75" align-y="2076" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>j</ml:id><ml:range><ml:real>0</ml:real><ml:real>40</ml:real></ml:range></ml:define></math><rendering item-idref="31" /></region><region region-id="1230" left="144" top="2067" width="36.75" height="12.75" align-x="152.25" align-y="2076" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>i</ml:id><ml:range><ml:real>0</ml:real><ml:real>40</ml:real></ml:range></ml:define></math><rendering item-idref="32" /></region><region region-id="1129" left="66" top="2090.25" width="53.25" height="27.75" align-x="78" align-y="2106" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:plus /><ml:real>-10</ml:real><ml:apply><ml:div /><ml:id>i</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:define></math><rendering item-idref="33" /></region><region region-id="1231" left="144" top="2097" width="27" height="18" align-x="157.5" align-y="2106" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply></ml:define></math><rendering item-idref="34" /></region><region region-id="1239" left="228" top="2097" width="60.75" height="18" align-x="249.75" align-y="2106" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:apply><ml:indexer /><ml:id>G</ml:id><ml:sequence><ml:id>i</ml:id><ml:id>j</ml:id></ml:sequence></ml:apply><ml:apply><ml:id>g</ml:id><ml:apply><ml:indexer /><ml:id>x</ml:id><ml:id>i</ml:id></ml:apply><ml:apply><ml:indexer /><ml:id>y</ml:id><ml:id>j</ml:id></ml:apply></ml:apply></ml:define></math><rendering item-idref="35" /></region><region region-id="1218" left="0" top="2124" width="6000" height="6" align-x="0" align-y="2124" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="1247" left="66" top="2138.25" width="164.25" height="62.25" align-x="135.75" align-y="2172" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>localmin</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>G</ml:id></ml:apply></ml:eval></math><rendering item-idref="36" /></region><region region-id="1229" left="294" top="2163" width="71.25" height="12.75" align-x="327" align-y="2172" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>min</ml:id><ml:id>G</ml:id></ml:apply></ml:eval></math><rendering item-idref="37" /></region><region region-id="1226" left="66" top="2223" width="147" height="12.75" align-x="136.5" align-y="2232" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>localmax</ml:id><ml:id>x</ml:id><ml:id>y</ml:id><ml:id>G</ml:id></ml:apply></ml:eval></math><rendering item-idref="38" /></region><region region-id="1228" left="294" top="2223" width="71.25" height="12.75" align-x="327.75" align-y="2232" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>max</ml:id><ml:id>G</ml:id></ml:apply></ml:eval></math><rendering item-idref="39" /></region><region region-id="1152" left="36" top="2258.25" width="321" height="48" align-x="36" align-y="2268" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If there is a known analytic function, you can use the Maximize or Minimize functions to find maxima or minima analytically. First, provide guess values to give the solver a place to start seeking a solution.</p></text></region><region region-id="790" left="66" top="2316.75" width="51.75" height="29.25" align-x="91.5" align-y="2334" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:matrix rows="2" cols="1"><ml:id>x</ml:id><ml:id>y</ml:id></ml:matrix><ml:matrix rows="2" cols="1"><ml:real>0</ml:real><ml:real>0</ml:real></ml:matrix></ml:define></math><rendering item-idref="40" /></region><region region-id="791" left="36" top="2360.25" width="326.25" height="24" align-x="45" align-y="2370" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Pass the name of the objective function and the guess values to Maximize.</p></text></region><region region-id="797" left="66" top="2403" width="115.5" height="36.75" align-x="106.5" align-y="2424" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:matrix rows="2" cols="1"><ml:id subscript="max">x</ml:id><ml:id subscript="max">y</ml:id></ml:matrix><ml:apply><ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false" /><ml:id>g</ml:id><ml:id>x</ml:id><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="41" /></region><region region-id="813" left="222" top="2415" width="48" height="16.5" align-x="246" align-y="2424" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id subscript="max">x</ml:id></ml:eval></math><rendering item-idref="42" /></region><region region-id="814" left="282" top="2415" width="63.75" height="16.5" align-x="307.5" align-y="2424" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:id subscript="max">y</ml:id></ml:eval></math><rendering item-idref="43" /></region><region region-id="799" left="36" top="2456.25" width="328.5" height="48" align-x="36" align-y="2466" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When optimizing a function of more than one parameter, Mathcad will return a vector of results. The first element in this vector corresponds to the first variable after the function name in the call to Maximize or Minimize (x in this case), and so on. </p></text></region><region region-id="815" left="66" top="2517" width="99" height="16.5" align-x="127.5" align-y="2526" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:eval><ml:apply><ml:id>g</ml:id><ml:id subscript="max">x</ml:id><ml:id subscript="max">y</ml:id></ml:apply></ml:eval></math><rendering item-idref="44" /></region><region region-id="818" left="66" top="2541" width="44.25" height="18" align-x="84" align-y="2550" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:real>0</ml:real></ml:apply><ml:id subscript="max">x</ml:id></ml:define></math><rendering item-idref="45" /></region><region region-id="822" left="144" top="2541" width="44.25" height="18" align-x="160.5" align-y="2550" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>Y</ml:id><ml:real>0</ml:real></ml:apply><ml:id subscript="max">y</ml:id></ml:define></math><rendering item-idref="46" /></region><region region-id="825" left="228" top="2541" width="80.25" height="18" align-x="244.5" align-y="2550" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:apply><ml:indexer /><ml:id>Z</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:id>g</ml:id><ml:id subscript="max">x</ml:id><ml:id subscript="max">y</ml:id></ml:apply></ml:define></math><rendering item-idref="47" /></region><region region-id="821" left="66" top="2580" width="217.5" height="204" align-x="174" align-y="2580" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="48" disable-calc="false"><inputs hide="false"><ml:sequence><ml:id>g</ml:id><ml:parens><ml:sequence><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>Z</ml:id></ml:sequence></ml:parens></ml:sequence></inputs><outputs hide="false" /></component><rendering item-idref="49" /></region><region region-id="1174" left="0" top="2790" width="6000" height="6" align-x="0" align-y="2790" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="853" left="36" top="2804.25" width="333.75" height="24" align-x="43.5" align-y="2814" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">You can also find maxima and minima in a constrained way using solve blocks. Provide guess values for any unknowns:</p></text></region><region region-id="846" left="66" top="2841" width="21.75" height="12.75" align-x="75.75" align-y="2850" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>x</ml:id><ml:real>2</ml:real></ml:define></math><rendering item-idref="50" /></region><region region-id="851" left="144" top="2841" width="23.25" height="12.75" align-x="155.25" align-y="2850" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:id>y</ml:id><ml:real>2</ml:real></ml:define></math><rendering item-idref="51" /></region><region region-id="849" left="36" top="2865" width="24" height="12.75" align-x="47.25" align-y="2874" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:id>Given</ml:id></math><rendering item-idref="52" /></region><region region-id="857" left="144" top="2864.25" width="159.75" height="12" align-x="154.5" align-y="2874" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Open the solve block with Given</p></text></region><region region-id="858" left="66" top="2889" width="36" height="12.75" align-x="90.75" align-y="2898" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:lessOrEqual /><ml:apply><ml:lessOrEqual /><ml:real>0</ml:real><ml:id>x</ml:id></ml:apply><ml:real>5</ml:real></ml:apply></math><rendering item-idref="53" /></region><region region-id="856" left="144" top="2888.25" width="215.25" height="48" align-x="159" align-y="2898" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Provide Boolean constraints. These say we want the minimum value for g in the space where x and y can be no greater than 5, nor less than 0.</p></text></region><region region-id="859" left="66" top="2907" width="37.5" height="12.75" align-x="92.25" align-y="2916" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:apply><ml:lessOrEqual /><ml:apply><ml:lessOrEqual /><ml:real>0</ml:real><ml:id>y</ml:id></ml:apply><ml:real>5</ml:real></ml:apply></math><rendering item-idref="54" /></region><region region-id="1237" left="228" top="2948.25" width="111.75" height="24" align-x="238.5" align-y="2958" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">Close with a call to <f family="Verdana" charset="0"><b><inlineAttr line-through="false"><c val="#000">Minimize</c></inlineAttr></b></f><sp /></p></text></region><region region-id="860" left="66" top="2943" width="114" height="36.75" align-x="105.75" align-y="2964" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:matrix rows="2" cols="1"><ml:id subscript="min">x</ml:id><ml:id subscript="min">y</ml:id></ml:matrix><ml:apply><ml:Minimize auto-method="true" method="conjugate" derivative-estimation="forward" variable-estimation="quadratic" linear-check="true" multistart="false" evolutionary="false" /><ml:id>g</ml:id><ml:id>x</ml:id><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="55" /></region><region region-id="864" left="66" top="2997" width="29.25" height="18" align-x="84" align-y="3006" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:real>0</ml:real></ml:apply><ml:id>x</ml:id></ml:define></math><rendering item-idref="56" /></region><region region-id="870" left="144" top="2997" width="29.25" height="18" align-x="160.5" align-y="3006" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>Y</ml:id><ml:real>0</ml:real></ml:apply><ml:id>y</ml:id></ml:define></math><rendering item-idref="57" /></region><region region-id="874" left="222" top="2997" width="50.25" height="18" align-x="238.5" align-y="3006" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>Z</ml:id><ml:real>0</ml:real></ml:apply><ml:apply><ml:id>g</ml:id><ml:id>x</ml:id><ml:id>y</ml:id></ml:apply></ml:define></math><rendering item-idref="58" /></region><region region-id="867" left="66" top="3021" width="43.5" height="18" align-x="84" align-y="3030" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>X</ml:id><ml:real>1</ml:real></ml:apply><ml:id subscript="min">x</ml:id></ml:define></math><rendering item-idref="59" /></region><region region-id="872" left="144" top="3021" width="43.5" height="18" align-x="160.5" align-y="3030" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>Y</ml:id><ml:real>1</ml:real></ml:apply><ml:id subscript="min">y</ml:id></ml:define></math><rendering item-idref="60" /></region><region region-id="875" left="222" top="3021" width="78.75" height="18" align-x="238.5" align-y="3030" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedUDScalar"><ml:apply><ml:indexer /><ml:id>Z</ml:id><ml:real>1</ml:real></ml:apply><ml:apply><ml:id>g</ml:id><ml:id subscript="min">x</ml:id><ml:id subscript="min">y</ml:id></ml:apply></ml:define></math><rendering item-idref="61" /></region><region region-id="1191" left="48" top="3048" width="217.5" height="204" align-x="156" align-y="3048" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="62" disable-calc="false"><inputs hide="false"><ml:sequence><ml:id>g</ml:id><ml:parens><ml:sequence><ml:id>X</ml:id><ml:id>Y</ml:id><ml:id>Z</ml:id></ml:sequence></ml:parens></ml:sequence></inputs><outputs hide="false" /></component><rendering item-idref="63" /></region><region region-id="1175" left="0" top="3258" width="6000" height="6" align-x="0" align-y="3258" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><pageBreak /></region><region region-id="905" left="36" top="3266.25" width="324" height="60" align-x="48" align-y="3276" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">When optimizing a function in Mathcad, guess values play an extremely important role in the solution which is found. Guesses far from the desired values may cause the solver to converge to a local maximum or minimum. For example, consider the function H, which has two peaks:</p></text></region><region region-id="1072" left="66" top="3333" width="199.5" height="18.75" align-x="100.5" align-y="3348" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define warning="WarnRedefinedBIUnit"><ml:function><ml:id>H</ml:id><ml:boundVars><ml:id>a</ml:id><ml:id>b</ml:id></ml:boundVars></ml:function><ml:apply><ml:minus /><ml:apply><ml:minus /><ml:apply><ml:plus /><ml:apply><ml:plus /><ml:apply><ml:minus /><ml:real>1000</ml:real><ml:apply><ml:mult style="auto-select" /><ml:real>3</ml:real><ml:apply><ml:pow /><ml:id>a</ml:id><ml:real>4</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:real>8</ml:real><ml:apply><ml:pow /><ml:id>a</ml:id><ml:real>3</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:real>66</ml:real><ml:apply><ml:pow /><ml:id>a</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:real>144</ml:real><ml:id>a</ml:id></ml:apply></ml:apply><ml:apply><ml:mult style="auto-select" /><ml:real>50</ml:real><ml:apply><ml:pow /><ml:id>b</ml:id><ml:real>2</ml:real></ml:apply></ml:apply></ml:apply></ml:define></math><rendering item-idref="64" /></region><region region-id="909" left="36" top="3372" width="268.5" height="189" align-x="169.5" align-y="3372" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><component item-idref="65" disable-calc="false"><inputs hide="false"><ml:id>H</ml:id></inputs><outputs hide="false" /></component><rendering item-idref="66" /></region><region region-id="910" left="36" top="3572.25" width="330" height="36" align-x="39" align-y="3582" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><text use-page-width="false" push-down="false" lock-width="true"><p style="Normal" margin-left="inherit" margin-right="inherit" text-indent="inherit" text-align="inherit" list-style-type="inherit" tabs="inherit">If you compare the results of Maximize with two different sets of guesses, you can see that one set of guesses finds the global maximum, while the other converges to the local maximum.</p></text></region><region region-id="1233" left="36" top="3627" width="29.25" height="12.75" align-x="46.5" align-y="3636" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>a</ml:id><ml:real>4.5</ml:real></ml:define></math><rendering item-idref="67" /></region><region region-id="1235" left="114" top="3627" width="30" height="12.75" align-x="125.25" align-y="3636" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b</ml:id><ml:real>3.5</ml:real></ml:define></math><rendering item-idref="68" /></region><region region-id="1078" left="222" top="3627" width="27" height="12.75" align-x="237" align-y="3636" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>a1</ml:id><ml:real>1</ml:real></ml:define></math><rendering item-idref="69" /></region><region region-id="934" left="288" top="3627" width="33" height="12.75" align-x="303.75" align-y="3636" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:id>b1</ml:id><ml:real>-4</ml:real></ml:define></math><rendering item-idref="70" /></region><region region-id="1234" left="36" top="3651" width="117.75" height="36.75" align-x="76.5" align-y="3672" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:matrix rows="2" cols="1"><ml:id subscript="max">a</ml:id><ml:id subscript="max">b</ml:id></ml:matrix><ml:apply><ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false" /><ml:id>H</ml:id><ml:id>a</ml:id><ml:id>b</ml:id></ml:apply></ml:define></math><rendering item-idref="71" /></region><region region-id="1079" left="222" top="3651" width="131.25" height="36.75" align-x="267" align-y="3672" show-border="false" show-highlight="false" is-protected="true" background-color="inherit" tag=""><math optimize="false" export="false" disable-calc="false"><ml:define><ml:matrix rows="2" cols="1"><ml:id subscript="max">a1</ml:id><ml:id subscript="max">b1</ml:id></ml:matrix><ml:apply><ml:Maximize auto-method="true" method="conjugate" 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