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: