Question

Find the extreme values of f on the region described by the inequality.

f(x, y) = e−xy; x^2 + 4y^2 ≤ 1

Answer 1

To find the extreme values of f on a region defined by an inequality, we first find the critical points of f by taking partial derivatives and setting them equal to zero. Then, we evaluate the function at the critical points and along the boundary of the region. Finally, we compare the values to find the extreme values.

Step-by-step explanation:

To find the extreme values of f on the region described by the inequality, we need to find the critical points of f inside the given region. First, let's find the critical points of f by taking the partial derivatives with respect to x and y, and setting them equal to zero. The partial derivative with respect to x is: fx(x, y) = -y*e^(-xy). The partial derivative with respect to y is: fy(x, y) = -x*e^(-xy). Setting both derivatives equal to zero gives us the critical point x = 0 and y = 0.

Next, we need to evaluate the function f at the critical points and along the boundary of the region. Since the given region is defined by the inequality x2 + 4y2 ≤ 1, we can substitute the boundary equation into the function f to find f along the boundary.

Finally, we compare the values of f at the critical points and along the boundary to find the extreme values of f.