Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$. (a) Find the values of $g ( 2 )$ and $g ( - 2 )$. (b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist. (c) Find the $x$-coordinate of each point at which the graph of $g$ has a horizontal tangent line. For each of these points, determine whether $g$ has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers. (d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.
Let $f$ be the continuous function defined on $[ - 4,3 ]$ whose graph, consisting of three line segments and a semicircle centered at the origin, is given above. Let $g$ be the function given by $g ( x ) = \int _ { 1 } ^ { x } f ( t ) d t$.
(a) Find the values of $g ( 2 )$ and $g ( - 2 )$.
(b) For each of $g ^ { \prime } ( - 3 )$ and $g ^ { \prime \prime } ( - 3 )$, find the value or state that it does not exist.
(c) Find the $x$-coordinate of each point at which the graph of $g$ has a horizontal tangent line. For each of these points, determine whether $g$ has a relative minimum, relative maximum, or neither a minimum nor a maximum at the point. Justify your answers.
(d) For $- 4 < x < 3$, find all values of $x$ for which the graph of $g$ has a point of inflection. Explain your reasoning.