Question

Find the absolute extrema of the given function on the indicated closed and bounded set . i ) (x, y) = xy − x − 3y; is the triangular region with vertices (0,0), (0,4), and (5,0).

Answer 1

To find the absolute extrema of the given function on the triangular region, determine the critical points and evaluate the function at the vertices. The absolute minimum is -13 and the absolute maximum is 12.

Step-by-step explanation:

To find the absolute extrema of the function f(x, y) = xy - x - 3y on the given triangular region, we need to evaluate the function at the critical points and the endpoints of the region.

Step 1: Find the critical points by taking the partial derivatives of f with respect to x and y, and setting them equal to zero.

Step 2: Evaluate f at the critical points and the vertices of the triangular region.

Step 3: Compare the function values to determine the absolute extrema.

In this case, the function has one critical point at (2, 2) and the vertices of the triangular region are (0, 0), (0, 4), and (5, 0). Evaluating the function at these points, we find that the absolute minimum is -13 at (2, 2) and the absolute maximum is 12 at (0, 4).