On the circle 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 all extreme values of the function 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Method of Lagrange MultipliersEnter objective function LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNiYtSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JC1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnL0Y5USdub3JtYWxGJ0YrRjs=use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
SetProperty(MathContainer0,value,NULL);
end module;
end use;
Enter constraints 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entered as functions 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use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
SetProperty(MathContainer1,value,NULL);
end module;
end use;
Enter coordinate variables, separated by commas:use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
SetProperty(MathContainer2,value,NULL);
end module;
end use;
JSFHuse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
_EnvDisableGlobalBindings:=true:
xxSOL := 1;
xxf := Do(%MathContainer0);
tempg := Do(%MathContainer1);
if nops([tempg])>1 then xxg := [tempg];
elif type(tempg,list) or type(tempg,set) then xxg := tempg;
else xxg := [tempg];
end if;
tempvar := op(Typesetting:-Parse(Typesetting:-FromMathML(DocumentTools:-GetProperty('MathContainer2','value'))))[2..-1];
if nops([tempvar])>1 then xxvar := [tempvar];
elif type(tempvar,list) or type(tempvar,set) then xxvar := tempvar;
else xxvar := [tempvar];
end if;
xxNC := nops(xxg);
_EnvExplicit := true;
xxSol := Student[MultivariateCalculus][LagrangeMultipliers](xxf,xxg,xxvar,output=detailed);
if xxSol=NULL then xxSOL := 0;
else xxSol1 := [xxSol];
xxN := nops(xxSol1);
if length(xxSol1)>500 then xxSol1:=evalf(xxSol1);end if;
xxSol1 := remove(has,xxSol1,I);
end if;
if xxSOL=0 then xxM := "Maple found no solutions" else
xxV := Vector([op(xxvar),seq(lambda[xxs],xxs=1..xxNC),xxf]):
xxV1 := Vector([op(xxvar),seq(lambda[xxs],xxs=1..xxNC),`f`]):
for j from 1 to xxN do
xxc||j := eval(xxV,xxSol1[j]);
od:
xxM := Matrix([xxV1,xxc||(1..xxN)]):
end if;
SetProperty(MathContainer3,value,MathML[Export](xxM)):
_EnvExplicit := false;
end module;
end use; use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
M := GetProperty(MathContainer3, expression);
if M<>NULL then
SetProperty(MathContainer3,value,MathML[Export](evalf[4](M))):
end if;
end module;
end use;
JSFHuse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
SetProperty(MathContainer3,value,NULL);
end module;
end use;
Tools\342\211\273Tasks\342\211\273Browse: Calculus - Differential\342\211\273Optimization\342\211\273Constrained OptimizationNote: Within the Optimization Assistant, it will be useful to make expeditious choices of Initial Values.
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LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JS1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi8lK2V4ZWN1dGFibGVHRjtGNkYrRk1GNg==Objective Function:LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1Constraint Equation or Sequence of Such: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 DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
Do(CO_1 = %MathContainer4);
Do(CO_2 = [%MathContainer5]);
if CO_1=NULL or CO_2=NULL then
CO_3 := Optimization[Interactive]()
else
CO_3 := Optimization[Interactive](CO_1,CO_2);
end if;
if CO_3 <> NULL then DocumentTools:-SetProperty('MathContainer6','value',CO_3); end if;
end module;
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
SetProperty('MathContainer4', 'value', NULL);
SetProperty('MathContainer5', 'value', NULL);
end module
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# GetProperty( component_name, attribute_name )
# and
# SetProperty( component_name, attribute_name, value )
# to access any of the attributes of the component.
# See ?CustomizingComponents for more information.
module()
SetProperty('MathContainer6', 'value', NULL);
end module;
end use;
LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1Result:LUknbXN0eWxlRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YnNictSSNtaUdGJDYjUSFGJy1JJW1yb3dHRiQ2JUYrLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRitGMkY1
JSFH
<Text-field style="Heading 1" layout="Heading 1">Solution from First Principles</Text-field>JSFH
Using Control-drag, enter the expression LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKCZtaW51cztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUScmIzk1NTtGJy9GMEY9RjktRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZXLUYsNiVRImdGJ0YvRjJGOQ==, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the left-hand side of the constraint equation.Context Menu: Assign to a Name\342\211\273FJSFHJSFHUsing the partial derivative operator in the Expression palette, formulate the sequence of equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRInhGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYtUSI9RidGPS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGSC8lKXN0cmV0Y2h5R0ZILyUqc3ltbWV0cmljR0ZILyUobGFyZ2VvcEdGSC8lLm1vdmFibGVsaW1pdHNHRkgvJSdhY2NlbnRHRkgvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZXLUkjbW5HRiQ2JEZBRj0tRkM2LVEiLEYnRj1GRi9GSkY0RktGTUZPRlFGUy9GVlEmMC4wZW1GJy9GWVEsMC4zMzMzMzMzZW1GJy1GLDYlRi4tRiM2JC1GLzYlUSJ5RidGMkY1Rj1GP0ZCRlpGZ24tRiw2JUYuLUYjNiUtRi82JVEnJiM5NTU7RicvRjNGSEY9RjJGNUY/RkJGWkY9.Press the Enter key.Context Menu: Solve\342\211\273SolveJSFHJSFH