Question

find the absolute maximum and minimum values of the function, subject to the given constraints k(x,y) = -x² - y² + 16x + 16y ; 0 ≤ x ≤ 9, y ≥ 0, and x+y ≤ 18

Answer 1

To find the absolute maximum and minimum values of the function, analyze the critical points and boundary points within the given constraints. The absolute maximum value is 0, occurring at (0, 0) and (9, 0), and the absolute minimum value is -324, occurring at (0, 18) and (9, 18).

Step-by-step explanation:

To find the absolute maximum and minimum values of the function, we need to analyze the critical points and boundary points within the given constraints. First, we calculate the partial derivatives of the function:

∂k/∂x = -2x + 16
∂k/∂y = -2y + 16

Setting both partial derivatives equal to zero, we find the critical point at (8, 8). Next, we check the boundary points:

For x = 0 and y = 0, the function evaluates to k(0, 0) = 0.
For x = 0 and y = 18, the function evaluates to k(0, 18) = -324.
For x = 9 and y = 0, the function evaluates to k(9, 0) = 0.
For x = 9 and y = 18, the function evaluates to k(9, 18) = -324.

Comparing these values, we find that the absolute maximum value of the function is 0, which occurs at (0, 0) and (9, 0), and the absolute minimum value is -324, which occurs at (0, 18) and (9, 18).