Answer:f(x, y) = e
−xy. For the interior of the region, we find the critical points:
fx = −ye−xy
, fy = −xe−xy, so the only critical point is (0, 0), and f(0, 0) = 1.
For the boundary, we use Lagrange multipliers.
g(x, y) = x
2 + 4y
2 = 1 ⇒ λ∇g = h2λx, 8λyi, so setting ∇f = λ∇g we get
−ye−xy = 2λx and −xe−xy = 8λy. The first of these gives e
−xy = −2λx/y,
and then the second gives −x(−2λx/y) = 8λy ⇒ x
2 = 4y
2
. Solving this
last equation with the constraint x
2 + 4y
2 = 1 gives x = ± √
1
2
and y = ±
1
2
√
2
.
Now f
± √
1
2
, ∓
1
2
√
2
= e
1/4 ≈ 1.284 and f
± √
1
2
, ±
1
2
√
2
= e
−1/4 ≈ 0.779.
The former are the maxima on the region and the latter are the minima.
Step-by-step explanation:
