Question

Find the absolute maximum and minimum values of f on the set d. f(x, y) = 6xy - x - 2y, where d is the closed triangular region with vertices (1, 0), (5, 0), and (1, 4).

Answer 1

To determine the absolute maximum and minimum of f(x, y) = 6xy - x - 2y on a given triangular region, compute the gradient for internal critical points, assess f on triangle edges, evaluate f at vertices, and compare these values.

Step-by-step explanation:

To find the absolute maximum and minimum values of the function f(x, y) = 6xy - x - 2y on the closed triangular region d with vertices (1, 0), (5, 0), and (1, 4), we must perform several steps:

1. Compute the gradient of f and find the critical points inside the region.
2. Examine the behavior of f on the boundaries of the region d.
3. Evaluate f at all found critical points and at the vertices of the triangle to determine the maximum and minimum values.

For the gradient of f, we get ∇f = (6y - 1, 6x - 2). Setting each component to zero, we have the system of equations 6y - 1 = 0 and 6x - 2 = 0, which yields the critical point (1/3, 1/6). However, this point does not lie within the triangular region.

Next, we check the three edges of the triangle by plugging in the respective constraints:

• For the edge from (1,0) to (5,0), y = 0 and f(x, 0) = -x on this edge.
• For the edge from (1,0) to (1,4), x = 1 and f(1, y) = 4y - 1 - 2y = 2y - 1 on this edge.
• For the edge from (1,4) to (5,0), the line can be represented by y = -x + 5, so f(x, -x + 5) is the function along this edge.

We also evaluate f at the vertices: f(1, 0) = -3, f(5, 0) = -5, and f(1, 4) = 14.

Lastly, by comparing all the values of f at the boundary and at the vertices, we can identify the absolute maximum and minimum of the function on the region d.