The graph of the function $f$ consists of three line segments. (a) Let $g$ be the function given by $g(x) = \int_{-4}^{x} f(t)\, dt$. For each of $g(-1)$, $g'(-1)$, and $g''(-1)$, find the value or state that it does not exist. (b) For the function $g$ defined in part (a), find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $-4 < x < 3$. Explain your reasoning. (c) Let $h$ be the function given by $h(x) = \int_{x}^{3} f(t)\, dt$. Find all values of $x$ in the closed interval $-4 \leq x \leq 3$ for which $h(x) = 0$. (d) For the function $h$ defined in part (c), find all intervals on which $h$ is decreasing. Explain your reasoning.
The graph of the function $f$ consists of three line segments.
(a) Let $g$ be the function given by $g(x) = \int_{-4}^{x} f(t)\, dt$. For each of $g(-1)$, $g'(-1)$, and $g''(-1)$, find the value or state that it does not exist.
(b) For the function $g$ defined in part (a), find the $x$-coordinate of each point of inflection of the graph of $g$ on the open interval $-4 < x < 3$. Explain your reasoning.
(c) Let $h$ be the function given by $h(x) = \int_{x}^{3} f(t)\, dt$. Find all values of $x$ in the closed interval $-4 \leq x \leq 3$ for which $h(x) = 0$.
(d) For the function $h$ defined in part (c), find all intervals on which $h$ is decreasing. Explain your reasoning.