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 $G(x) = \int_{0}^{x} f(t)\, dt$. (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 $\lim_{x \rightarrow 2} \frac{G(x)}{x^{2} - 2x}$. (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 $G(x) = \int_{0}^{x} f(t)\, dt$.\\
(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 $\lim_{x \rightarrow 2} \frac{G(x)}{x^{2} - 2x}$.\\
(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.