Calc III · §14.7 extrema on a closed region (disk)

Extreme values of f = x² + 2y² on the disk

Domain x² + y² ≤ 1. Same plan as always: interior critical points, then the boundary — but the boundary is now a circle, so we ride around it (or use Lagrange multipliers) instead of walking straight edges. Compare the candidates to pick the max and min.

The bowl f over the disk — rim = boundary
Map view — level ellipses meet the circle (Lagrange)
f around the circle — g(t) = f(cos t, sin t) = 1 + sin²t
Candidate pointcomes fromf valuerunning

Interior criticals + the boundary circle → compare the list. 🔵