A continuous function $f$ is defined on the closed interval $- 4 \leq x \leq 6$. The graph of $f$ consists of a line segment and a curve that is tangent to the $x$-axis at $x = 3$, as shown in the figure above. On the interval $0 < x < 6$, the function $f$ is twice differentiable, with $f ^ { \prime \prime } ( x ) > 0$.
(a) Is $f$ differentiable at $x = 0$ ? Use the definition of the derivative with one-sided limits to justify your answer.
(b) For how many values of $a , - 4 \leq a < 6$, is the average rate of change of $f$ on the interval $[ a , 6 ]$ equal to 0 ? Give a reason for your answer.
(c) Is there a value of $a , - 4 \leq a < 6$, for which the Mean Value Theorem, applied to the interval $[ a , 6 ]$, guarantees a value $c , a < c < 6$, at which $f ^ { \prime } ( c ) = \frac { 1 } { 3 }$ ? Justify your answer.
(d) The function $g$ is defined by $g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t$ for $- 4 \leq x \leq 6$. On what intervals contained in $[ - 4,6 ]$ is the graph of $g$ concave up? Explain your reasoning.