Consider the function F(x,y) = 1 - x3 - y2 + x3y2. Consider the curve C given parametrically as x(t) = t1/3, y(t) = t1/2 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.
Fx = -3x2 + 3x2y2
Fy = -2y + x3*2y
Fx(t1/3,t1/2) = -3t2/3 + 3t5/3 = 1/(3t2/3) * λ
Fy(t1/3,t1/2) = -2t1/2 + 2t3/2 = 1/(2t1/2) * λ
9t4/9 * (-1 + t) = λ
4t1/4 * (-1 + t) = λ
9t4/9 = 4t1/4
t7/36 = 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?