Chapter 14.7, Problem 25E

To determine

That the function $f\left(x,y\right)=\frac{1}{3}{x}^3+\frac{2}{3}{y}^{3/2}-xy$ satisfies $f\left(x,y\right)\ge 0\left(x\ge 0,y\ge 0\right)$ and find the critical points of the function.

(a) We need to verify that the set of critical points of $f$ is a parabola $y=x^2$ and that the second derivative test fails for these points.

(b) We need to show that for fixed $b$, the function $g\left(x\right)=f\left(x,b\right)$ is concave up for $x>0$ with a critical point at $x=b^{1/2}$.

(c) We need to conclude that $f\left(a,b\right)\ge f\left(b^{1/2},b\right)=0$ for all $a,b\ge 0$