The problem asks to "Select all values of $x$ where the function $f$ has a critical point such that $f'(x) = 0$."

To find where the derivative $f'(x) = 0$, we need to look at points where the slope of the tangent line to the curve is horizontal (i.e., where the graph has a local maximum, minimum, or a point of inflection with a horizontal tangent).

Looking at the graph, critical points occur where the slope is zero (i.e., flat spots on the curve):

• $x = -6$ (local minimum)
• $x = -2$ (local maximum)
• $x = 3$ (local minimum)
• $x = 6$ (local maximum)

These are the points where $f'(x) = 0$. So the correct values of $x$ are:

• $x = -6$
• $x = -2$
• $x = 3$
• $x = 6$

Would you like further clarification on how to identify critical points or explore other related concepts?

Here are 5 related questions to consider:

1. How do you find critical points from the derivative of a function?
2. What is the significance of $f'(x) = 0$ in calculus?
3. How can you determine if a critical point is a local minimum or maximum?
4. What happens when $f'(x)$ does not exist at a point but $f(x)$ is continuous?
5. How does the second derivative test help classify critical points?

Tip: Critical points occur where the derivative of a function is zero or undefined, and they are essential for finding local extrema (maximums and minimums).