What you "need to know" is how to find max and min!
Whoever gave you this problem clearly expects you to know that if a max or min occurs in the interior of a region in two dimensions the two partial derivatives must both be 0. Of course, the max and/or min might be on a boundary. The boundary consists of the three line segments bounding the triangle. Write each with y as a function of x. If a max or min occurs in the interior of a segment, must be where the derivative of y with respect to x 0. Finally, a max or min may occur at the boundary of an interval and those boundaries are the three vertices of the triangle.