To determine how many local extrema (maxima or minima) the function $f(x)$ has, we can analyze the graph of its derivative $f'(x)$.

Local extrema in $f(x)$ occur where $f'(x)$ changes sign:

• From positive to negative: This indicates a local maximum in $f(x)$.
• From negative to positive: This indicates a local minimum in $f(x)$.

Step-by-Step Analysis:

1. The curve of $f'(x)$ starts below the x-axis (negative) and rises to cross the x-axis. This indicates a change from negative to positive, corresponding to a local minimum in $f(x)$.
2. The curve then peaks and crosses the x-axis downward, changing from positive to negative. This indicates a local maximum in $f(x)$.
3. It then continues downward, crossing the x-axis again. This change from negative to positive suggests another local minimum in $f(x)$.
4. Finally, the curve peaks again and crosses the x-axis downwards, which corresponds to another local maximum in $f(x)$.

Conclusion:

The derivative $f'(x)$ changes sign four times, indicating there are four local extrema in $f(x)$ (two maxima and two minima).

Answer: 4

Would you like further details or have any questions?

Here are five related questions you might find useful:

1. How can you determine the exact location of local maxima and minima using the derivative?
2. What is the significance of the second derivative in finding local extrema?
3. How do inflection points relate to the first derivative?
4. How can you determine if a function is concave up or concave down using its derivative?
5. What are the implications of a derivative being zero over an interval?

Tip: When analyzing a derivative graph, the points where the derivative crosses the x-axis are key indicators of possible local maxima or minima.