Let $f$ be a function defined on the closed interval $[0,7]$. The graph of $f$, consisting of four line segments, is shown above. Let $g$ be the function given by $g(x) = \int_{2}^{x} f(t)\, dt$. (a) Find $g(3)$, $g'(3)$, and $g''(3)$. (b) Find the average rate of change of $g$ on the interval $0 \leq x \leq 3$. (c) For how many values $c$, where $0 < c < 3$, is $g'(c)$ equal to the average rate found in part (b)? Explain your reasoning. (d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the interval $0 < x < 7$. Justify your answer.
Let $f$ be a function defined on the closed interval $[0,7]$. The graph of $f$, consisting of four line segments, is shown above. Let $g$ be the function given by $g(x) = \int_{2}^{x} f(t)\, dt$.
(a) Find $g(3)$, $g'(3)$, and $g''(3)$.
(b) Find the average rate of change of $g$ on the interval $0 \leq x \leq 3$.
(c) For how many values $c$, where $0 < c < 3$, is $g'(c)$ equal to the average rate found in part (b)? Explain your reasoning.
(d) Find the $x$-coordinate of each point of inflection of the graph of $g$ on the interval $0 < x < 7$. Justify your answer.