This problem asks us to analyze the graph of $g'(x)$, the first derivative of $g(x)$, to find the critical points and determine the nature of $g(x)$ (local maximum, minimum, or neither) at each of these critical values.

Step-by-step approach:

1. Identify Critical Points:

   • Critical points occur where $g'(x) = 0$ or where $g'(x)$ is undefined. In this case, look for:
     • Where the graph of $g'(x)$ crosses the x-axis (i.e., $g'(x) = 0$).
     • Any discontinuities in the graph of $g'(x)$.

2. Examine the Graph:

   • The graph of $g'(x)$ shows three key points where $g'(x) = 0$ (where the graph intersects the x-axis):
     • At $x = -2$
     • At $x = 1$
     • At $x = 4$
   • There is also a discontinuity at $x = 0$, where the graph "jumps."

3. Use the First Derivative Test:

   • To determine the nature of each critical point (local maximum, minimum, or neither), check how $g'(x)$ changes sign around each critical point:
     • At $x = -2$: $g'(x)$ changes from positive (to the left of $x = -2$) to negative (to the right of $x = -2$). This indicates a local maximum at $x = -2$.
     • At $x = 1$: $g'(x)$ changes from negative to positive, indicating a local minimum at $x = 1$.
     • At $x = 4$: $g'(x)$ does not change sign (it remains positive), so this is neither a maximum nor a minimum.
     • At $x = 0$: There's a discontinuity. Since we cannot determine the behavior of $g'(x)$ at this point using the first derivative test, this point requires special attention. Given the nature of $g'(x)$, $g(x)$ may not have an extremum at $x = 0$.

Conclusion:

• Local maximum at $x = -2$.
• Local minimum at $x = 1$.
• Neither at $x = 4$ and $x = 0$.

Would you like further explanation or details on any of these steps?

Related questions:

1. What is the significance of the first derivative in identifying local extrema?
2. How does the second derivative test differ from the first derivative test?
3. Why is $g(x)$ continuous even though $g'(x)$ is not at certain points?
4. Can we always use the first derivative test for non-continuous derivative graphs?
5. What other methods can be used to confirm extrema besides the first derivative test?

Tip:

When analyzing the first derivative graph, always check for sign changes around critical points, as they provide key information about the behavior of the original function.