Answer
a) Saddle point.
b) Local maximum point.
c) Test is inconclusive
Work Step by Step
We are given the second partial derivatives $g_{xx}$, $g_{xy}$, and $g_{yy}$. Therefore, using these values, we can calculate D for the Second Derivatives Test to determine if $g(0,2)$ is a local minimum, local maximum, or saddle point.
a) $D=(-1)(1)-(6)^2<0$. Therefore, $g(0,2)$ is a saddle point.
b) $D=(-1)(-8)-(2)^2>0$ and $g_{xx}<0$, so this is a local maximum point.
c) $D=(4)(9)-(6)^2=0$, which means the test is inconclusive and the point can be any of the three choices.
