Question
Consider the graph of  
f
′
(
x
)
  below. How many local maxima does  
f
(
x
)
  have?
A curve on a coordinate plane.
A coordinate plane has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 4 to 4 in increments of 1. From left to right, a curve labeled f prime rises in quadrant 3 to a local maximum at left-parenthesis negative 2 comma 0 right-parenthesis, falls to a local minimum at left-parenthesis negative 1.5 comma negative 3.2 right-parenthesis, rises to a local maximum at left-parenthesis 0 comma 0 right-parenthesis, and falls to left-parenthesis 0.8 comma negative 0.4 right-parenthesis, and then rises passing through left-parenthesis 1 comma 0 right-parenthesis. All coordinates are approximate.

Select the correct answer below:


0
 
1
 
2
 
3
 
4
 



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.

Consider the graph of f'(x) below. How many local maxima does f(x) have? A curve on a coordinate plane. A coordinate plane has a horizontal x-axis labeled from negative 5 to 5 in increments of 1 and a vertical y-axis labeled from negative 4 to 4 in increments of 1. From left to right, a curve labeled f prime rises in quadrant 3 to a local maximum at (−2, 0), falls to a local minimum at (−1.5, −3.2), rises to a local maximum at (0, 0), and falls to (0.8, −0.4), and then rises passing through (1, 0). All coordinates are approximate.

Learn how to determine the number of local maxima of a function f(x) by analyzing the graph of its derivative f'(x). This problem involves key calculus concepts such as derivatives, critical points, and the first derivative test, suitable for high school students in Grades 11-12.

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