Question

Find the absolute maximum and minimum values of f on the set D.

f(x, y) = x² + y² + x²y + 7, D = ≤ 1

Answer 1

To find the absolute maximum and minimum values of f on the set D, we consider the critical points and the boundary of D. The critical point is (0, 0) and the absolute maximum value is 8 at (0, 0). The absolute minimum value is 6 at (-1, -1).

Step-by-step explanation:

To find the absolute maximum and minimum values of f on the set D, we need to consider the critical points and the boundary of D. Let's start by finding the critical points by taking the partial derivatives of f with respect to x and y, and setting them equal to zero:

1. ∂f/∂x = 2x + 2xy = 0
2. ∂f/∂y = 2y + x² = 0

Solving these two equations simultaneously, we find that the critical point is (0, 0). Next, we need to check the values of f at the boundary of D: when x = 1, x = -1, y = 1, and y = -1.

To determine the absolute maximum and minimum values, we evaluate f at these critical points and boundary points. After calculations, we find that the absolute maximum value of f is 8 and it occurs at the critical point (0, 0). The absolute minimum value of f is 6 and it occurs at the boundary point (-1, -1).