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The graph provided is of the second derivative \( f''(x) \) of a function \( f(x) \). To determine the inflection points of \( f(x) \), we need to identify where \( f''(x) \) changes sign, as these points correspond to where the concavity of \( f(x) \) changes.

Looking at the graph of \( f''(x) \), the x-coordinates where \( f''(x) = 0 \) (the curve crosses the x-axis) are potential candidates for inflection points, provided there is a sign change on either side of those points.

From the graph:
- \( f''(x) = 0 \) at \( x \approx 3 \) and \( x \approx 6 \).

To confirm these are inflection points, we observe the behavior of \( f''(x) \):
- At \( x = 3 \), \( f''(x) \) changes from negative to positive, indicating an inflection point.
- At \( x = 6 \), \( f''(x) \) changes from positive to negative, also indicating an inflection point.

Thus, the x-coordinates of the inflection points of \( f(x) \) are approximately:

\[
x = 3, 6
\]

Would you like further details or clarifications on this process?

Here are 5 follow-up questions to expand on this topic:
1. What is the significance of an inflection point in the context of concavity?
2. How does the second derivative test help in identifying local maxima and minima?
3. Can there be more inflection points if the second derivative graph touches but doesn’t cross the x-axis?
4. How does the behavior of the second derivative relate to the slope of the first derivative?
5. What happens at an inflection point in terms of the function's first derivative?

**Tip:** Always check for a sign change in \( f''(x) \) around points where \( f''(x) = 0 \), as a zero value without a sign change does not indicate an inflection point.

The graph of the second derivative f''(x) of a function f(x) is shown. State the x-coordinates of the inflection points of f.

Learn how to find the x-coordinates of inflection points by analyzing the second derivative graph. This example highlights the concavity change condition, focusing on when f''(x) crosses the x-axis. Suitable for students in Grades 11-12.

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