Answer to Question #247211 in Calculus for Susan

Question #247211

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).


1
Expert's answer
2021-10-12T09:40:16-0400

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


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