calculus on manifolds (spivak) problem 2-41 (c)

Question

2-41. let $f:\Bbb{R} \times \Bbb{R} \rightarrow \Bbb{R}$ be differentiable. for each $x \in \Bbb{R}$ define $g_x: \Bbb{R} \rightarrow \Bbb{R}$ by $g_x(y)=f(x,y)$. suppose that for each $x$ there is a unique $y$ with $g'_x(y)=0$; let $c(x)$ be this $y$.

(a) If $D_{2,2}f(x,y) \neq 0$ for all $(x,y)$, show that $c$ is differentiable and $c'(x)=-\frac{D_{2,1}f(x,c(x))}{D_{2,2}f(x,c(x))}$.

Hint: $g'_x(y)=0$ can be written $D_2f(x,y)=0$.

(b) show that if $c'(x)=0$, then for some $y$ we have

$D_{2,1}f(x,y)=0$

$D_2f(x,y)=0$

(c) let $f(x,y)=x(y \,log \,y \,- \,y)- y \,log\, x$. find $\underset{\frac{1}{2}\leq x \leq 2}{max}(\underset {\frac{1}{2}\leq y \leq 1}{min}f(x,y))$.

for the given $f(x,y)$, $c(x)=e^{log\,x/x}. \, c'(x)=0 \implies x=1$ and since $c(1)=1,$ according to part (b), $D_{2,2}f(1,1)=0$ and $D_2f(1,1)=0$

But how do I know $\underset{\frac{1}{2}\leq x \leq 2}{max}(\underset {\frac{1}{2}\leq y \leq 1}{min}f(x,y))$ occurrs at $(1,1)$? What other critical points do I have to look at?

please give a hint.please do not provide solution.