The continuous function $f$ is defined on the interval $- 4 \leq x \leq 3$. The graph of $f$ consists of two quarter circles and one line segment, as shown in the figure above. Let $g ( x ) = 2 x + \int _ { 0 } ^ { x } f ( t ) d t$. (a) Find $g ( - 3 )$. Find $g ^ { \prime } ( x )$ and evaluate $g ^ { \prime } ( - 3 )$. (b) Determine the $x$-coordinate of the point at which $g$ has an absolute maximum on the interval $- 4 \leq x \leq 3$. Justify your answer. (c) Find all values of $x$ on the interval $- 4 < x < 3$ for which the graph of $g$ has a point of inflection. Give a reason for your answer. (d) Find the average rate of change of $f$ on the interval $- 4 \leq x \leq 3$. There is no point $c , - 4 < c < 3$, for which $f ^ { \prime } ( c )$ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.
The continuous function $f$ is defined on the interval $- 4 \leq x \leq 3$. The graph of $f$ consists of two quarter circles and one line segment, as shown in the figure above. Let $g ( x ) = 2 x + \int _ { 0 } ^ { x } f ( t ) d t$.
(a) Find $g ( - 3 )$. Find $g ^ { \prime } ( x )$ and evaluate $g ^ { \prime } ( - 3 )$.
(b) Determine the $x$-coordinate of the point at which $g$ has an absolute maximum on the interval $- 4 \leq x \leq 3$. Justify your answer.
(c) Find all values of $x$ on the interval $- 4 < x < 3$ for which the graph of $g$ has a point of inflection. Give a reason for your answer.
(d) Find the average rate of change of $f$ on the interval $- 4 \leq x \leq 3$. There is no point $c , - 4 < c < 3$, for which $f ^ { \prime } ( c )$ is equal to that average rate of change. Explain why this statement does not contradict the Mean Value Theorem.