Let $f$ be a function defined on the closed interval $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above. (a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer. (b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer. (c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning. (d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.
& - 1 & 3 & 6 & 7
Let $f$ be a function defined on the closed interval $- 5 \leq x \leq 5$ with $f ( 1 ) = 3$. The graph of $f ^ { \prime }$, the derivative of $f$, consists of two semicircles and two line segments, as shown above.
(a) For $- 5 < x < 5$, find all values $x$ at which $f$ has a relative maximum. Justify your answer.
(b) For $- 5 < x < 5$, find all values $x$ at which the graph of $f$ has a point of inflection. Justify your answer.
(c) Find all intervals on which the graph of $f$ is concave up and also has positive slope. Explain your reasoning.
(d) Find the absolute minimum value of $f ( x )$ over the closed interval $- 5 \leq x \leq 5$. Explain your reasoning.