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MathGroup Archive 2006

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Re: Tricky visualization of maximization problem

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71007] Re: [mg71004] Tricky visualization of maximization problem
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sat, 4 Nov 2006 23:07:01 -0500 (EST)
  • Reply-to: hanlonr at cox.net

expr=x1^2 + 4*x1*x2 + 3*x2^2;

sub1=x2->Sqrt[1-x1^2];

f1[x1_] = Simplify[expr/.sub1];

pts1={x1,f1[x1]}/.Solve[f1'[x1]==0,x1]//
    Simplify

{{-Sqrt[(1/10)*(5 + Sqrt[5])], 2 - Sqrt[5]}, {Sqrt[(1/10)*(5 - Sqrt[5])], 2 + Sqrt[5]}}

Plot[f1[x1],{x1,-1,1},Epilog->
      {Red,AbsolutePointSize[5],Point/@pts1}];

pt3D1=({x1,x2,expr}/.sub1)/.
      {x1->pts1[[2,1]]}//Simplify

{Sqrt[(1/10)*(5 - Sqrt[5])], Sqrt[(1/10)*(5 + Sqrt[5])], 2 + Sqrt[5]}

sub2=x2->-Sqrt[1-x1^2];

f2[x1_] = Simplify[expr/.sub2];

pts2={x1,f2[x1]}/.Solve[f2'[x1]==0,x1]//
    Simplify

{{Sqrt[(1/10)*(5 + Sqrt[5])], 2 - Sqrt[5]}, {-Sqrt[(1/10)*(5 - Sqrt[5])], 2 + Sqrt[5]}}

Plot[f2[x1],{x1,-1,1},Epilog->
      {Red,AbsolutePointSize[5],Point/@pts2}];

pt3D2=({x1,x2,expr}/.sub2)/.
      {x1->pts2[[2,1]]}//Simplify

{-Sqrt[(1/10)*(5 - Sqrt[5])], -Sqrt[(1/10)*(5 + Sqrt[5])], 2 + Sqrt[5]}

Needs["Graphics`"];

DisplayTogether[
    Plot3D[expr*Boole[x1^2+x2^2<1],
      {x1,-1,1},{x2,-1,1},
      Mesh->False,PlotPoints->125],
    Show[Graphics3D[{Red,AbsolutePointSize[6],
          Point/@{pt3D1,pt3D2}}]]];


Bob Hanlon

---- Uwe Ziegenhagen <newsgroup at ziegenhagen.info> wrote: 
> Hi,
> 
> I want to maximize
> 
> x1^2 + 4*x1*x2 + 3*x2^2  (eq.1)
> 
> under the constraint
> 
> x1^2 + x2^2 == 1 (eq. 2)
> 
> So far no problem, Maximize gives me 2 + sqrt(5)
> 
> But how can I display this visually?
> 
> For eq. 1 I can use Plot3D[], for eq. 2 ImplicitPlot[] but how to have
> them in one picture?
> 
> 
> Thanks in advance,
> 
> 
> Uwe
> 

--

Bob Hanlon
hanlonr at cox.net



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