Question

Find the absolute maximum and minimum values of the function f(x,y) = x + y - xy

on the closed triangular region with vertices (0,0), (0,2), and (4,0).

Answer 1

$\displaystyle \text{First, we find the derivative with respect to x and y}\\ \frac{\partial f(x,y)}{\partial x}=1-y=0 \implies y =1\\ \frac{\partial f(x,y)}{\partial y} = 1 -x=0 \implies x =1\\ f(1,1)=1\\ \text{(1,1) is the point inside a given region. Hence minima is at the (1,1) that is 1}\\ \text{Now we calculate the maxima by substituting all the vertices in the function one by one}\\ f(0,0)=0\\ f(0,2)=0\\ f(4,0)=4\\ \text{Therefore the maximum value is 4 at vertex (0,4)}$