Question

1. Suppose that f(x,y)=xy. 

1. Find the maximum value of the function if x and y are constrained to sum to 1.

        b) How can you be sure this is a maximum and not a minimum?

Answer 1

Since we must have x+y=1, then we can solve for, say, y terms of x using that equation

This gives y=1-x

Which we then substitute into f to get f(x,y)=xy=x(1-x)=x-x²

This is now a function of x alone, so we now just have to maximize the function f⁢(x)=10⁢x-x2" role="presentation" style="display: inline; line-height: normal; font-size: medium; overflow-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: "Times New Roman"; position: relative;">f(x)=x−x2on the interval [0,1]

f'(x)=1-2x=0

x=1/2

f''(x)=-2<0 

Then the Second Derivative Test tells us that x=1is a local maximum for f,

 and hence x=1 must be the global maximum on the interval [0,1] 

(since f=0" role="presentation" style="display: inline; line-height: normal; font-size: medium; overflow-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: "Times New Roman"; position: relative;">f=0 t the endpoints of the interval). So since y=1-x=1-1/2=1/2

f=0" role="presentation" style="display: inline; line-height: normal; font-size: medium; overflow-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: "Times New Roman"; position: relative;">

f=0" role="presentation" style="display: inline; line-height: normal; font-size: medium; overflow-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: "Times New Roman"; position: relative;">Then the maximum area occurs for a rectangle whose width and height both are

f=0" role="presentation" style="display: inline; line-height: normal; font-size: medium; overflow-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: "Times New Roman"; position: relative;">1/2.

f=0" role="presentation" style="display: inline; line-height: normal; font-size: medium; overflow-wrap: normal; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: "Times New Roman"; position: relative;">