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Extrema problem

  1. Jul 17, 2007 #1
    1. The problem statement, all variables and given/known data

    Find and classify the local and absolute extrema of the function
    [tex]f(x,y)=x-xy[/tex] over the region
    [tex]D={(x,y)|x^2+y^2\leq1 and x+y\geq0}[/tex]

    2. Relevant equations



    3. The attempt at a solution
    Critical points are where the first derivative (gradient) is 0.
    [tex]\nablaf=(1-y, -x)=0[/tex]
    So critical point a=(0,1)

    In order to classify the critical point, find the Hessian matrix of f at a:
    [tex]H=\left(\begin{array}{cc}0&-1\\-1&0\end{array}\right)[/tex]
    Then the quadratic form is:
    [tex]Q(x,y)=Hk\cdotk=-2xy[/tex]
    [tex]Q(a)=Q(0,1)=0[/tex]

    Which means the test is inconclusive??? I.e., the critical point a is a saddle point, which is neither local maximum or minimum. Am I right?


    Also, how do you find the absolute extreme of the function on the region aforementioned?

    I tried to convert x^2+y^2<=1 to polar coordinates, which gives
    [tex]r^2\cos^2\eta+r^2\sin^2\eta\leq1[/tex]
    [tex]r^2\leq1[/tex]
    [tex]0<r\leq1[/tex]
    However, this doesn't help much as the original function converted to polar coordinates is not straightforward to find its extrema given the domain of r and theta.
     
  2. jcsd
  3. Jul 17, 2007 #2
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