The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$. (a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ? (b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) d x$. (c) For $- 5 < x < 6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer. (d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.
The graph of the continuous function $g$, the derivative of the function $f$, is shown above. The function $g$ is piecewise linear for $- 5 \leq x < 3$, and $g ( x ) = 2 ( x - 4 ) ^ { 2 }$ for $3 \leq x \leq 6$.
(a) If $f ( 1 ) = 3$, what is the value of $f ( - 5 )$ ?
(b) Evaluate $\int _ { 1 } ^ { 6 } g ( x ) d x$.
(c) For $- 5 < x < 6$, on what open intervals, if any, is the graph of $f$ both increasing and concave up? Give a reason for your answer.
(d) Find the $x$-coordinate of each point of inflection of the graph of $f$. Give a reason for your answer.