Answer
$f(x,y) =4xy$ has a maximum value subject to the constraint $x + y = 16$ at the point (8,8)
Work Step by Step
$f(x,y)=4xy$
The constraint becomes $x+y-16=0$ with $g(x,y)=x+y-16$
$F(x,y,\lambda)=f(x,y)-\lambda.g(x,y)=4xy-\lambda(x+y-16)=4xy-\lambda x -\lambda y + 16\lambda$
$F_{x}(x,y,\lambda)=4y-\lambda$
$F_{y}(x,y,\lambda)=4x-\lambda$
$F_{\lambda}(x,y,\lambda)=-x-y+16$
$(1) 4y-\lambda =0 \rightarrow \lambda =4y$
$(2) 4x-\lambda =0 \rightarrow \lambda =4x$
$(3) -x-y+16=0$
Set the expressions equal:
$4y=4x$
$x=y$
Substitute for x=y in Equation (3)
$-y-y+16=0$
$y=8 \rightarrow x=8$
The value of is f(8,8)=256
Let x=8.1 and y=7.9; then f(8.1;7.9)=255.96
Because a nearby point has a value smaller than 256, the value 256 is probably not a minimum.
