Find the absolute maxima and minima of the function on the given domain.
-$$f ( x , y ) = 5 x y ^ { 2 } + 8 x y \quad \text { on the trapezoidal region with vertices } ( 0,0 ) , ( 1,0 ) , ( 0,2 ) \text {, and } ( 1,1 )$$
A) Absolute maximum: 16 at (0, 2); absolute minimum: 0 at (0, 0)
B) Absolute maximum: 10 at (2, 0); absolute minimum: 5 at (1, 0)
C) Absolute maximum: 13 at (1, 1); absolute minimum: 5 at (1, 0)
D) Absolute maximum: 13 at (1, 1); absolute minimum: 0 at (0, 0)

Question

Quizplus · Accepted Answer

To find the absolute maxima and minima of the function $f(x, y) = 5xy^2 + 8xy$ on the given trapezoidal region, we evaluate the function at the vertices of the region and any critical points within the region. The vertices are (0,0), (1,0), (0,2), and (1,1). Evaluating the function at these points gives:- $f(0,0) = 5(0)(0)^2 + 8(0)(0) = 0$- $f(1,0) = 5(1)(0)^2 + 8(1)(0) = 0$- $f(0,2) = 5(0)(2)^2 + 8(0)(2) = 0$- $f(1,1) = 5(1)(1)^2 + 8(1)(1) = 13$Since the function is linear in both $x$ and $y$, and there are no other points in the interior of the region where the derivative is zero (critical points), the maximum and minimum values must occur at the vertices. The maximum value is 13 at (1,1), and the minimum value is 0 at (0,0), (1,0), and (0,2), but we only need to mention one of these points for the minimum.

Find the Absolute Maxima and Minima of the Function on the Given