Problem

Suppose that $f(x,y)$ is a differentiable function of $(x,y)$ for $(x,y)\in\mathbb{R}^2$, and $f_{xx}$, $f_{xy}$, and $f_{yy}$ are continuous on $\{(x,y):-2<x<2\text{ and }-1<y<1\}$. Suppose that

$$f_x(1,0)=0=f_y(1,0),\qquad f_{xx}(1,0)=4,\qquad f_{xy}(1,0)=f_{yx}(1,0)=3,\qquad f_{yy}(1,0)=2.$$

(a) (10 pts) Let $g(t)=f(e^{3t},\sin(t))$ for $t\in\mathbb{R}$. Does $g$ have a relative minimum at $0$? Justify your answer.

(b) (10 pts) Does $f$ have a relative minimum at $(1,0)$? Justify your answer.