Q:

Extrema of Real-Valued Functions

Let $ f(x,y) = 1 + xy + x - 2y $ and let $ D $ be the trangular region in $ \Bbb R^2 $ with vertices $ (1, -2) $ , $ (5, -2), $ and $ (2, 1) $. Find the absolute maximum and minimum values of $ f $ on $ D $. Give all the points where these extreme values occur.

A:

Welcome to the site. Normally you should include your own thoughts on the problem, and tell us where you are stuck. I will give you some general pointers for your problem.
We have $f_x(x,y) = y+1$ and $f_y(x,y) = x-2$. These two derivatives are simultaneously $0$ at $(x,y) = (2,-1)$. Check if this point is inside of your triangle.
You also need to check the values of your function on the boundary of the triangle. For instance, the edge connecting $(1,-2)$ to $(5,-2)$ is part of the line $y=-2$ in the $(x,y)$-plane. Plug this into your function and see what it looks like: $f(x,-2) = 1-2x+x+4 = 5-x$. Then check where this function has its maximum on this particular side of the triangle. Do this for all three sides.

