Chapter 9.3, Problem 30E

Let $f\left(x,y\right)=y^2-2x^{2}y+4x^3+20x^2$. The only critical points are $\left(-2,4\right)$, $\left(0,0\right)$, and $\left(5,25\right)$. Which of the following correctly describes the behavior of f at these points? Source: Society of Actuaries.

a. $\left(-2,4\right)$: local (relative) minimum

$\left(0,0\right)$: local (relative) minimum

$\left(5,25\right)$: local (relative) maximum

b. $\left(-2,4\right)$: local (relative) minimum

$\left(0,0\right)$: local (relative) maximum

$\left(5,25\right)$: local (relative) maximum

c. $\left(-2,4\right)$: neither a local (relative) minimum nor local (relative) maximum

$\left(0,0\right)$: local (relative) maximum

$\left(5,25\right)$: local (relative) minimum

d. $\left(-2,4\right)$: local (relative) maximum

$\left(0,0\right)$: neither a local (relative) minimum nor local (relative) maximum

$\left(5,25\right)$: local (relative) minimum

e. $\left(-2,4\right)$: neither a local (relative) minimum nor local (relative) maximum

$\left(0,0\right)$: local (relative) minimum

$\left(5,25\right)$: neither a local (relative) minimum nor local (relative) maximum