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## Homework Statement

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.

## 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?