Given $f(x,y) = x^2+y^2$ and the constraint $x^4+y^4=16$, find the maximum and minimum values. Using lagrange multipliers, I found the relationship y=±x and the critical points $(8^.25, 8^.25), (8^.25, -8^.25), (-8^.25, 8^.25), (-8^.25, -8^.25) = 2\sqrt{8}$. However, there must be another set of critical points and this is especially noticeable when graphed.
Am I not seeing another relationship between $f$ and the constraint? I read online that now all extrema can be found my lagrange multipliers, so is that the case?
Thank you!