## Calculus with Applications (10th Edition)

With $x=8, y=16, f(x,y) =3xy^{2}$ is maximized at 6144
We are given $f(x,y)=3xy^{2}$ with $x+y=24$ so $y=24-x$ $f(x,y)=3x(24-x)^{2}=3x(576-48x+x^{2})=1728x-144x^{2}+3x^{3}$ Now differentiate f(x) with respect to x and equate it to '0' as shown below $f_{x}(x,y)=1728-288x+9x^{2} = 0$ $x=24 \rightarrow y=0 \rightarrow$ The value of $3xy^{2}$ is 0 or $x=8 \rightarrow y=16 \rightarrow$ The value of $3xy^{2}$ is 6144 Since $6144\gt0$, with $x=8, y=16, f(x,y) =3xy^{2}$ is maximized at 6144