Optimization in two variables and Lagrange multipliers practice question 6

The wire is a closed, bounded curve, so its coldest and hottest points both exist, and Lagrange multipliers find candidates for each. What must you do with the candidates before answering?

With $g(x, y) = x^{2} + y^{2}$, the conditions $T_x = \lambda g_x$ and $T_y = \lambda g_y$ read $2 = 2\lambda x$ and $-1 = 2\lambda y$, together with $x^{2} + y^{2} = 20$.

Dividing the first condition by the second gives $x = -2y$. Substitute into the circle to get two candidate points, then compare $T$ at both.