Let $g$ be a continuous function with $g ( 2 ) = 5$. The graph of the piecewise-linear function $g ^ { \prime }$, the derivative of $g$, is shown above for $- 3 \leq x \leq 7$. (a) Find the $x$-coordinate of all points of inflection of the graph of $y = g ( x )$ for $- 3 < x < 7$. Justify your answer. (b) Find the absolute maximum value of $g$ on the interval $- 3 \leq x \leq 7$. Justify your answer. (c) Find the average rate of change of $g ( x )$ on the interval $- 3 \leq x \leq 7$. (d) Find the average rate of change of $g ^ { \prime } ( x )$ on the interval $- 3 \leq x \leq 7$. Does the Mean Value Theorem applied on the interval $- 3 \leq x \leq 7$ guarantee a value of $c$, for $- 3 < c < 7$, such that $g ^ { \prime \prime } ( c )$ is equal to this average rate of change? Why or why not?
Let $g$ be a continuous function with $g ( 2 ) = 5$. The graph of the piecewise-linear function $g ^ { \prime }$, the derivative of $g$, is shown above for $- 3 \leq x \leq 7$.
(a) Find the $x$-coordinate of all points of inflection of the graph of $y = g ( x )$ for $- 3 < x < 7$. Justify your answer.
(b) Find the absolute maximum value of $g$ on the interval $- 3 \leq x \leq 7$. Justify your answer.
(c) Find the average rate of change of $g ( x )$ on the interval $- 3 \leq x \leq 7$.
(d) Find the average rate of change of $g ^ { \prime } ( x )$ on the interval $- 3 \leq x \leq 7$. Does the Mean Value Theorem applied on the interval $- 3 \leq x \leq 7$ guarantee a value of $c$, for $- 3 < c < 7$, such that $g ^ { \prime \prime } ( c )$ is equal to this average rate of change? Why or why not?