The graph of the differentiable function $y = f ( x )$ with domain $0 \leq x \leq 10$ is shown in the figure above. The area of the region enclosed between the graph of $f$ and the $x$-axis for $0 \leq x \leq 5$ is 10 , and the area of the region enclosed between the graph of $f$ and the $x$-axis for $5 \leq x \leq 10$ is 27 . The arc length for the portion of the graph of $f$ between $x = 0$ and $x = 5$ is 11, and the arc length for the portion of the graph of $f$ between $x = 5$ and $x = 10$ is 18 . The function $f$ has exactly two critical points that are located at $x = 3$ and $x = 8$.
(a) Find the average value of $f$ on the interval $0 \leq x \leq 5$.
(b) Evaluate $\int _ { 0 } ^ { 10 } ( 3 f ( x ) + 2 ) d x$. Show the computations that lead to your answer.
(c) Let $g ( x ) = \int _ { 5 } ^ { x } f ( t ) d t$. On what intervals, if any, is the graph of $g$ both concave up and decreasing? Explain your reasoning.
(d) The function $h$ is defined by $h ( x ) = 2 f \left( \frac { x } { 2 } \right)$. The derivative of $h$ is $h ^ { \prime } ( x ) = f ^ { \prime } \left( \frac { x } { 2 } \right)$. Find the arc length of the graph of $y = h ( x )$ from $x = 0$ to $x = 20$.