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Math Help - Finding absolute extremaa

  1. #1
    Senior Member Pinkk's Avatar
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    Finding absolute extremaa

    Let f(x,y)=x^{3}-3x-y^{3}+12y. Find the absolute maximum and absolute minimum of f on the region D=\{(x,y)|x^{2}+y^{2}\leq 1\}

    So I tried to find critical points, and the only ones are (\pm 1,\pm 2), which are not in the region D, so now I have to find the extrema on the boundary. This is where I am having much trouble. I tried doing both f(x,\sqrt{1-x^{2}}) and f(\sqrt{1-y^{2}},y), and I end with pretty nasty equations that seem impossible (or at least extremely difficult) if I set their derivatives equal to 0. Any tips/work shown would be greatly appreciated.
    Last edited by Pinkk; March 24th 2009 at 06:56 PM.
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  2. #2
    Senior Member Pinkk's Avatar
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    No takers? I tried using the substitution x=cos(t),y=sin(t) but it was still an extremely convoluted equation.
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  3. #3
    Senior Member Pinkk's Avatar
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    So, I redid trig substitution and eventually ended up with g'(\theta)=3(sin^{3}\theta + cos^{3}\theta +3cos\theta). I'm stuck on how to solve for g'(\theta)=0
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  4. #4
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    Quote Originally Posted by Pinkk View Post

    So, I redid trig substitution and eventually ended up with g'(\theta)=3(sin^{3}\theta + cos^{3}\theta +3cos\theta). I'm stuck on how to solve for g'(\theta)=0
    if you let \tan \theta = t, then g'(\theta)=0 will give you t^3 + 3t^2 + 4=0, which has 3 real distinct solutions none of which are nice numbers!
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