Calculus: Max.+Min of f(x,y)=16x^2+17y^2 on the disk D: x^2+y^2≤1?

First, find the critical points inside D.
f_x = 32x, f_y = 34y.

Setting these equal to 0 yields (x, y) = (0, 0), which is inside D.
Note that f(0, 0) = 0, which is clearly the minimum value of f (via sum of squares).
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Next, check the boundary of D.
We can use Lagrange Multipliers on this, but better yet is to parameterize x^2 + y^2 = 1
via x = cos t, y = sin t.

f(cos t, sin t) = 16 cos^2(t) + 17 sin^2(t)
...................= 16 + sin^2(t), via cos^2(t) + sin^2(t) = 1.

This clearly takes on its maximal value when sin^2(t) = 1 (like when t = π/2 or 3π/2).
==> The maximum value is 16 + 1 = 17.

I hope this helps!