Find the extreme values of f on the region described by the inequality. f(x, y) = e−xy; x2 + 4y2 ≤ 1

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:

playboycarti! *

she tryna be crate.

describing distributions is one of the key skills you’ll need to earn a high score on the ap statistics exam. if you need proof of this, just flip through some past exam questions, which can be found at the collegeboard website. you’ll notice that the first free response question is almost always a question requiring you to look at a graph and describe it.

step-by-step explanation: just trying to get points sorry : (