Final answer:
The absolute maximum of the function f(x, y) = 3xy - x - 2y on the closed triangular region is 3 at the vertex (1, 4), and the absolute minimum is -5 at the vertex (5, 0).
Step-by-step explanation:
To find the absolute maximum and minimum values of the function f(x, y) = 3xy - x - 2y on the closed triangular region with vertices (1, 0), (5, 0), and (1, 4).
We need to evaluate the function at the vertices and along the edges of the triangle, as well as any critical points within the region.
First, evaluate f at the vertices:
- f(1, 0) = 3(1)(0) - 1 - 2(0) = -1
- f(5, 0) = 3(5)(0) - 5 - 2(0) = -5
- f(1, 4) = 3(1)(4) - 1 - 2(4) = 3
Next, check the edges of the triangular region. The edges are the lines connecting the vertices, and to evaluate along them.
We'd set up equations for these lines (y=0 for the base, x=1 for the left edge, and the line connecting (1,4) and (5,0) for the right edge) and plug these into the function to see if any other extrema exist.
Additionally, we should check for any critical points by finding the gradient of f, setting it to zero, and solving for x and y within the region.
However, in this case, the critical points fall outside the triangular region, so they are not considered.
The maximum value of f on this set is then 3, which occurs at the vertex (1, 4), while the minimum value of f is -5, which occurs at the vertex (5, 0).