Question

Find the absolute maximum and minimum values of f on the set D. f(x, y)= xy2, D = x >= 0, y >= 0, x2 + y2 <= 3

Maximum value =
Minimum value =

Answer 1

To find the absolute maximum and minimum values of the function f(x, y) = xy^2 over the set D, where D is constrained by non-negative x, y and x^2 + y^2 ≤ 3, we must find the critical points and evaluate f on the boundary of D.

Step-by-step explanation:

The question asks to find the absolute maximum and minimum values of the function f(x, y) = xy2 on the set D, where D is defined by x ≥ 0, y ≥ 0, and x2 + y2 ≤ 3. To find these values, one must consider both the interior and the boundary of D. Critical points inside D are found where the gradient of f is zero and, on the boundary of D, by studying the function along the curve defined by x2 + y2 = 3.

To find critical points within the interior of D, we take the partial derivatives of f with respect to x and y and set them to zero. The boundary is examined using the constraint to parameterize x or y and then optimizing the resulting single-variable function. Since both x and y are non-negative, the minimum value of the function is at (0, 0), which gives f(0, 0) = 0. The absolute maximum occurs on the boundary of D since increasing both x and y increases f, and the largest value can be found where x2 + y2 = 3. Therefore, one has to use calculus to maximize f with respect to the constraint.

.