Question

How do you minimize and maximize `f(x,y)=x^3-y^2-xy` constrained to `xy=4`?

Answer 1

Substitute `y=4/x`.

Explanation:

Substitute `y=4/x`. Remember that `x != 0`.

So `f` now is univariate and looks like

`f(x) = x^3 -16/x^2 - 4`
graph{x^3 -16/x^2 - 4 [-10, 10, -500, 500]}
Now optimize it by finding the critical points.

`f'(x) = 3x^2 + 32/x^3`

`f'(-2/root(5)3) = 0`

`x = -2/root(5)3` is a local maximum.