Answer
$f(x,y)=2xy+4$ has a minimum value subject to the constraint $x+y=20$; it is at the point (10,10).
The value of (10,10) is 204
Work Step by Step
$f(x,y)=2xy+4$
The constraint becomes $x+y-20=0$ with $g(x,y)=x+y-20$
$F(x,y,\lambda)=f(x,y)-\lambda.g(x,y)=2xy+4-\lambda(x+y-20)=2xy+4-\lambda x -\lambda y + 20\lambda$
$F_{x}(x,y,\lambda)=2y-\lambda$
$F_{y}(x,y,\lambda)=2x-\lambda$
$F_{\lambda}(x,y,\lambda)=-x-y+20$
$(1) 2y-\lambda =0 \rightarrow \lambda =2y$
$(2) 2x-\lambda =0 \rightarrow \lambda =2x$
$(3) -x-y+20=0$
Set the expressions equal:
$2y=2x$
$x=y$
Substitute for x=y in Equation (3):
$-y-y+20=0$
$y=108 \rightarrow x=10$
The value of is f(10,10)=204
Let x=10.1 and y=9.9; then, f(10.1;9.9)=203.98
Because a nearby point has a value smaller than 204, the value 204 is probably not a minimum.
f(x,y)=2xy+4 has a minimum value subject to the constraint x+y=20; it is at the point (10,10). The value of (10,10) is 204.
