(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider the function F(x,y) = 1 - x^{3}- y^{2}+ x^{3}y^{2}. Consider the curve C given parametrically as x(t) = t^{1/3}, y(t) = t^{1/2}for t ≥ 0. Determine the minimum and maximum of F(x,y) along the curve C.

3. The attempt at a solution

I think this is basically a max/min problem with a constraint function, so I will try to use Lagrange multipliers.

F_{x}= -3x^{2}+ 3x^{2}y^{2}

F_{y}= -2y + x^{3}*2y

F_{x}(t^{1/3},t^{1/2}) = -3t^{2/3}+ 3t^{5/3}= 1/(3t^{2/3}) * λ

F_{y}(t^{1/3},t^{1/2}) = -2t^{1/2}+ 2t^{3/2}= 1/(2t^{1/2}) * λ

9t^{4/9}* (-1 + t) = λ

4t^{1/4}* (-1 + t) = λ

9t^{4/9}= 4t^{1/4}

t^{7/36}= 4/9

t = (4/9)^{36/7}

And at this point I'm thinking there's no way this problem is this disgusting. Ideas on where I went wrong?

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# Homework Help: Finding Max/Min Values on Parametric Functions

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