The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, for $- 7 \leq x \leq 7$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = - 3 , x = 2$, and $x = 5$, and a vertical tangent line at $x = 3$. (a) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative minimum. Justify your answer. (b) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative maximum. Justify your answer. (c) Find all values of $x$, for $- 7 < x < 7$, at which $f ^ { \prime \prime } ( x ) < 0$. (d) At what value of $x$, for $- 7 \leq x \leq 7$, does $f$ attain its absolute maximum? Justify your answer.
The figure above shows the graph of $f ^ { \prime }$, the derivative of the function $f$, for $- 7 \leq x \leq 7$. The graph of $f ^ { \prime }$ has horizontal tangent lines at $x = - 3 , x = 2$, and $x = 5$, and a vertical tangent line at $x = 3$.\\
(a) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative minimum. Justify your answer.\\
(b) Find all values of $x$, for $- 7 < x < 7$, at which $f$ attains a relative maximum. Justify your answer.\\
(c) Find all values of $x$, for $- 7 < x < 7$, at which $f ^ { \prime \prime } ( x ) < 0$.\\
(d) At what value of $x$, for $- 7 \leq x \leq 7$, does $f$ attain its absolute maximum? Justify your answer.