5. (12 points) Let $g$ be continuous on $[0, 2]$ and differentiable on $(0, 2)$. Also assume that $g(0) = 1$, $g(1) = 1$, and $g(2) = 2$. Prove that there exists a number $c \in (0, 2)$ such that $g(c) = \frac{1}{9}$.
EXTRA CREDIT: Which number is larger, $4^{4.000000000000001}$ or $4.000000000000001^4$? Or are they equal?
Prove your assertion.