To determine the number of local maxima of $f(x)$, we need to analyze the graph of its derivative $f'(x)$.

Key Points:

• Local maxima of $f(x)$ occur where $f'(x)$ changes from positive to negative.
• This means that $f'(x)$ should be zero (i.e., $f'(x) = 0$) and the slope changes from increasing to decreasing.

Analysis of the Graph:

• The graph of $f'(x)$ reaches zero at $x = -2$, $x = 0$, and $x = 1$.
• At $x = -2$: $f'(x)$ changes from positive to negative (rising to falling), indicating a local maximum of $f(x)$.
• At $x = 0$: $f'(x)$ does not change signs (positive to negative) before and after $x = 0$; instead, it rises, reaches a maximum, and then falls but does not cross the x-axis. So, no local maximum of $f(x)$ here.
• At $x = 1$: $f'(x)$ also does not cross the x-axis from positive to negative, so it doesn’t indicate a local maximum of $f(x)$.

Conclusion:

There is only one local maximum for $f(x)$, occurring at $x = -2$.

The correct answer is:

1

Would you like further details or have any questions?

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Here are 5 related questions to consider:

1. What does a local minimum of $f(x)$ correspond to on the graph of $f'(x)$?
2. How would the graph of $f''(x)$ help in identifying concavity of $f(x)$?
3. What if the graph of $f'(x)$ was shifted vertically? How would that affect the local maxima of $f(x)$?
4. How can you determine points of inflection for $f(x)$ using $f'(x)$?
5. What would happen if $f'(x)$ had a point where it was zero but didn’t change sign?

Tip: Remember, the behavior of $f(x)$ is directly tied to the sign and changes in $f'(x)$. Analyzing the derivative carefully can help you understand the original function's behavior.