Have you met Lagrange Multipliers? That's one way.
You could go to the trouble to substitute and work the problem in one variable, but that might be a bit messy.
f(x,y) = x^2 + x(y^2 - 1)
and definition limit x^2 + y^2 = or < 17?
I know you use partial derivates to get the extreme points of f(x,y), but I don't understand how to solve for the extreme points on the definition limit circel. Someone could explain maybe, thanks?
Fine. Substituting for y^2 actually looks pretty straight-forward. Let's see what you get.
Note: The direct substitution deal only with the EQUALITY of the expression. You must then think about the interior. Are there extrema on the inside of the structure? How can you tell?