Let's analyze the problem based on the image provided.

(a) Identifying points where $f'$ is not defined:

The graph consists of straight line segments joined at various points, indicating sharp changes in direction at some points. Derivatives are not defined at points where there are corners or cusps because the slope changes abruptly. In this case, we need to identify where the function has corners.

Looking at the graph:

• At $x = -2$: The graph changes direction sharply (corner).
• At $x = 1$: Another sharp change in direction occurs (corner).
• At $x = 4$: A final sharp corner occurs.

Thus, the derivative $f'$ is not defined at $x = -2$, $x = 1$, and $x = 4$.

Answer to (a):

Option C is correct: " $f'$ is not defined at $x = -2$, $x = 1$, and $x = 4$ because at these points the graph has corners."

(b) Graphing the derivative of $f$:

Since the original graph consists of linear segments, the derivative of each linear segment will be a constant value, as the slope remains the same within each segment. Let's analyze the slopes of each segment:

1. From $x = -2$ to $x = 0$, the slope is: $\frac{2 - 0}{0 - (-2)} = \frac{2}{2} = 1$ The derivative here is $1$.

2. From $x = 0$ to $x = 1$, the slope is: $\frac{-2 - 2}{1 - 0} = \frac{-4}{1} = -4$ The derivative here is $-4$.

3. From $x = 1$ to $x = 4$, the slope is: $\frac{2 - (-2)}{4 - 1} = \frac{4}{3}$ The derivative here is $\frac{4}{3}$.

4. From $x = 4$ to $x = 6$, the slope is: $\frac{2 - 4}{6 - 4} = \frac{-2}{2} = -1$ The derivative here is $-1$.

Thus, the graph of the derivative would have:

• A value of $1$ from $x = -2$ to $x = 0$,
• A value of $-4$ from $x = 0$ to $x = 1$,
• A value of $\frac{4}{3}$ from $x = 1$ to $x = 4$,
• A value of $-1$ from $x = 4$ to $x = 6$.

The derivative is undefined at the points $x = -2$, $x = 1$, and $x = 4$, which correspond to the sharp corners where the slope changes abruptly.

Would you like a more detailed explanation or further assistance with graphing? Let me know if you have any questions.

Here are 5 related questions for further understanding:

1. How do you determine the derivative of a piecewise linear function?
2. What is the significance of corners or cusps in a graph regarding the derivative?
3. Why is the derivative not defined at certain points like sharp corners?
4. How do the slopes of each linear segment relate to the graph of the derivative?
5. What happens to the derivative at points where the graph has discontinuities?

Tip:

Remember, corners indicate places where the derivative is not defined because the direction of the slope changes abruptly, resulting in no single tangent line.