https://file.aiquickdraw.com/mathbot/file/1726152461656fozqtwdz.jpg

### Part b. Values of \( x \in (-3, 3) \) where \( g(x) \) is not differentiable:

To determine where \( g(x) \) is not differentiable, we look for points where:
1. There is a sharp corner or cusp.
2. The tangent is vertical (undefined slope).
3. The function is discontinuous (but that's already covered in part a).

From the graph:
- At \( x = 0 \), there is a sharp point where the curve changes direction sharply.
  
Thus, **\( g(x) \) is not differentiable at \( x = 0 \)**.

### Part c. Sketch of \( g'(x) \) (graph of the derivative of \( g(x) \)):

To sketch the derivative \( g'(x) \), observe the slopes of the graph of \( g(x) \) at different intervals:

1. For \( x < 0 \) (approximately from \( x = -3 \) to \( x = 0 \)), the function is decreasing, meaning \( g'(x) < 0 \). It has a negative slope that seems constant.
2. At \( x = 0 \), there is a sharp point, implying that the derivative does not exist there.
3. For \( x > 0 \) (from \( x = 0 \) to approximately \( x = 3 \)), the function is increasing initially, so \( g'(x) > 0 \), and the slope appears to gradually decrease to zero.

The sketch would reflect these slope changes:
- Negative constant slope for \( x < 0 \).
- Undefined derivative at \( x = 0 \) (a gap in the graph).
- Positive decreasing slope after \( x = 0 \) for a short interval and eventually approaching zero.

Let me know if you'd like more detail on how to sketch the graph of \( g'(x) \) based on these observations, or if you have any further questions!

### 5 Related Questions:
1. How do you determine points of non-differentiability from a graph?
2. Why is the derivative undefined at a sharp corner?
3. What is the relationship between a function’s increasing/decreasing behavior and its derivative?
4. Can a function be continuous but not differentiable? Provide examples.
5. How can we estimate the derivative at various points of a graph?

### Tip:
To sketch the derivative graph, focus on the slopes of the original function. The steeper the slope of the function, the greater the magnitude of its derivative.

Use the graph of g in the figure to do the following:
a. Find the values of x in (–3,3) at which g is not continuous.
b. Find the values of x in (–3,3) at which g is not differentiable.
c. Sketch a graph of g'.

Analyze the graph of g(x) to determine where the function is not continuous and not differentiable in the interval (–3,3). Explore key concepts such as continuity, differentiability, and how to sketch the derivative graph g'.

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