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How many inflection points does the function *f* (*x*) have?

 2 3 4 5

To find the number of inflection points for the function \( f(x) \), we need to analyze where the concavity of the graph changes — that is, where the function transitions from concave up (shaped like a "U") to concave down (shaped like an "n") or vice versa.

Looking at the provided graph, inflection points occur where the curve changes from concave up to concave down or from concave down to concave up. These are the points where the second derivative of the function, \( f''(x) \), changes sign.

Here’s the visual analysis of the graph:

- Between \( x = -5 \) and \( x = -4 \), the function is concave down.
- Between \( x = -4 \) and \( x = -1 \), the function is concave up.
- Between \( x = -1 \) and \( x = 1.5 \), the function is concave down.
- Between \( x = 1.5 \) and \( x = 3.5 \), the function is concave up.

Each change in concavity marks an inflection point.

From this, we can identify **three inflection points** in the graph where the concavity changes.

Thus, the correct answer is **3**.

Would you like more details on how inflection points are calculated analytically, or do you have other questions?

Here are 5 follow-up questions:

1. What is the mathematical definition of an inflection point?
2. How can we find inflection points by solving \( f''(x) = 0 \)?
3. How do first and second derivatives relate to concavity?
4. How would we calculate inflection points for a specific polynomial function?
5. What role do critical points play in determining the shape of a graph?

**Tip**: To find inflection points more precisely, you can compute the second derivative of a function and check where it changes sign.

How many inflection points does the function f(x) have?

Analyze the number of inflection points in the graph of the function f(x). This calculus problem involves determining changes in concavity by inspecting the graph visually or using the second derivative test. Ideal for advanced high school calculus students.

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