Let's analyze and solve the given problem step by step. Here’s the process:

The graph of $f(x)$:

• It consists of line segments, a semicircle, and another set of line segments.
• $g(x)$ is defined as: $g(x) = \int_0^x f(t) \, dt$ This means $g'(x) = f(x)$, by the Fundamental Theorem of Calculus.

Question (b): Evaluate $f'(3)$ and $f'(6)$:

• $f'(3)$: At $x = 3$, the graph shows a change in direction of the semicircle. $f'(3)$ is undefined because $f(x)$ is not smooth (it has a cusp at $x = 3$).

• $f'(6)$: At $x = 6$, the graph of $f(x)$ is a horizontal line, meaning $f'(6) = 0$.

Question (c): Evaluate $g'(3)$ and $g'(6)$:

• Since $g'(x) = f(x)$, we simply evaluate $f(x)$ at $x = 3$ and $x = 6$:
  • $g'(3) = f(3)$: From the graph, $f(3) = -1$.
  • $g'(6) = f(6)$: From the graph, $f(6) = 0$.

Question (d): Local maximum of $g(x)$ on $(0, 6)$:

• For $g(x)$ to have a local maximum, $g'(x) = f(x)$ must change sign from positive to negative.
• From the graph, $f(x)$ changes from positive to negative at $x = 2$, so $g(x)$ has a local maximum at $x = 2$.

Question (e): Local minimum of $g(x)$ on $(0, 6)$:

• For $g(x)$ to have a local minimum, $g'(x) = f(x)$ must change sign from negative to positive.
• From the graph, $f(x)$ changes from negative to positive at $x = 4$, so $g(x)$ has a local minimum at $x = 4$.

Final Answers:

(b)
$f'(3) = \text{undefined}, \ f'(6) = 0$

(c)
$g'(3) = -1, \ g'(6) = 0$

(d)
Local maximum of $g(x)$ occurs at $x = 2$.

(e)
Local minimum of $g(x)$ occurs at $x = 4$.

Would you like me to elaborate on any part of this solution? Here are five related questions for deeper understanding:

1. Why is $f'(3)$ undefined at $x = 3$, and how do cusps affect derivatives?
2. How does the Fundamental Theorem of Calculus relate $g(x)$ and $f(x)$?
3. What are the conditions for local extrema in integrals of functions?
4. Can $g(x)$ have a local extremum at a point where $f(x)$ is undefined?
5. How does the semicircle segment contribute to the behavior of $g(x)$?

Tip: For interpreting problems involving $g(x) = \int_0^x f(t) \, dt$, always start by analyzing the graph of $f(x)$ for critical points and changes in sign.