Answer
Function $f(x,y)=x^{4}y^{2}$ subject to the constraint $g(x,y)=2x+4y-60$ has a maximum value of 4000000 when x=20, y=5.
Work Step by Step
We are given $f(x,y)=x^{4}y^{2}$
where a unit of x cost is $\$2$, a unit of cost y is $\$4$ and $\$60$ is available, so that the combination of x and y is represented by the point $(x,y)=2x+4y=60 \rightarrow g(x,y)=2x+4y-60$
$F(x,y,\lambda)=f(x,y)-\lambda.g(x,y)=x^{4}y^{2}-\lambda(2x+4y-60)=x^{4}y^{2}-2\lambda x - 4\lambda y + 60\lambda$
$F_{x}(x,y,\lambda)=4x^{3}y^{2}-2\lambda$
$F_{y}(x,y,\lambda)=2x^{4}y-4\lambda$
$F_{\lambda}(x,y,\lambda)=-2x-4y+60$
$(1) 4x^{3}y^{2}-2\lambda=0 \rightarrow \lambda =2x^{3}y^{2}$
$(2) 2x^{4}y-4\lambda =0 \rightarrow \lambda =\frac{x^{4}y}{2}$
$(3) -2x-4y+60=0$
Set the expressions equal: $2x^{3}y^{2}=\frac{x^{4}y}{2}$
$4x^{3}y^{2}=x^{4}y$
$x=4y$
Substitute for x=4y in Equation (3)
$-2(4y)-4y+60=0$
$y=5 \rightarrow x=20$
The value of is $f(20,5)=4000000$
Let x=19 and y=5.5 then $f(19,5.5)=3942210$. Because a nearby point has a value smaller than 4000000, the value 4000000 is probably not a minimum.
Function $f(x,y)=x^{4}y^{2}$ subject to constraint $g(x,y)=2x+4y-60$ has a maximum value of 4000000 when x=20, y=5.
