https://file.aiquickdraw.com/mathbot/file/17277147034658kvsdcgy.jpg

Let's analyze the image, which appears to show a question related to the function \( f(x) = |x| \).

### (a) Sketch the graph of the function:
The correct graph for \( f(x) = |x| \) is a "V" shape, where:
- For \( x \geq 0 \), \( f(x) = x \) (a straight line with slope 1).
- For \( x < 0 \), \( f(x) = -x \) (a straight line with slope -1).

In the image, the correct graph is the fourth one on the far right (which resembles a "V"). 

### (b) For what values of \( x \) is \( f \) differentiable?
The function \( f(x) = |x| \) is differentiable everywhere except at \( x = 0 \), where there's a sharp corner. Hence, the function is differentiable for \( x \neq 0 \).

The correct option should be:
- **\( x \neq 0 \)**, not all real \( x \) as indicated in the image.

### (c) Find a formula for \( f'(x) \):
The derivative of \( f(x) = |x| \) can be expressed as a piecewise function:
\[
f'(x) = \begin{cases} 
1 & \text{if } x > 0, \\
-1 & \text{if } x < 0, \\
\text{undefined} & \text{at } x = 0.
\end{cases}
\]

Let me know if you'd like further details or clarifications! Here are some related questions for practice:

1. How would the graph of \( g(x) = -|x| \) differ from \( f(x) = |x| \)?
2. Why is \( f(x) = |x| \) not differentiable at \( x = 0 \)?
3. Can you describe the behavior of the second derivative \( f''(x) \) for \( f(x) = |x| \)?
4. How would you evaluate \( f'(x) \) using the limit definition of the derivative for \( f(x) = |x| \)?
5. What is the integral of \( f(x) = |x| \) over the interval \( [-2, 2] \)?

**Tip**: When determining differentiability, check for corners or cusps in the graph!

Let f(x) = |x|. (a) Sketch the graph of the function. (b) For what values of x is f differentiable? (c) Find a formula for f'(x).

Explore the differentiability of the absolute value function f(x) = |x|. This problem examines the graph, points of differentiability, and the derivative formula for this piecewise function. Ideal for high school calculus students.

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