Let $f$ be a continuous function defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$. (a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer. (b) Let $P$ be the function defined by $P ( x ) = G ( x ) \cdot f ( x )$. Find $P ^ { \prime } ( 3 )$. (c) Find $\lim _ { x \rightarrow 2 } \frac { G ( x ) } { x ^ { 2 } - 2 x }$. (d) Find the average rate of change of $G$ on the interval $[ - 4,2 ]$. Does the Mean Value Theorem guarantee a value $c , - 4 < c < 2$, for which $G ^ { \prime } ( c )$ is equal to this average rate of change? Justify your answer.
Let $f$ be a continuous function defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$, consisting of four line segments, is shown above. Let $G$ be the function defined by $G ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$.\\
(a) On what open intervals is the graph of $G$ concave up? Give a reason for your answer.\\
(b) Let $P$ be the function defined by $P ( x ) = G ( x ) \cdot f ( x )$. Find $P ^ { \prime } ( 3 )$.\\
(c) Find $\lim _ { x \rightarrow 2 } \frac { G ( x ) } { x ^ { 2 } - 2 x }$.\\
(d) Find the average rate of change of $G$ on the interval $[ - 4,2 ]$. Does the Mean Value Theorem guarantee a value $c , - 4 < c < 2$, for which $G ^ { \prime } ( c )$ is equal to this average rate of change? Justify your answer.