Exercise 10

Suppose that $g$ is a differentiable function and $g'(2) = 0$. In addition, suppose that on $1 \lt x\lt 2$ and $2 \lt x \lt 3$ it is known that $g'(x)$ is positive.

1. Does $g$ have a local maximum, local minimum, or neither at $x = 2$? Why?

2. Suppose that $g''(x)$ exists for every $x$ such that $1 \lt x \lt 3$. Reasoning graphically, describe the behavior of $g''(x)$ for $x$-values near $2$.

3. Besides being a critical number of $g$, what is special about the value $x = 2$ in terms of the behavior of the graph of $g$?