Let's solve this problem by analyzing the second derivative $f''(x) = x^2 (x - 2)^4 (x - 3)^2$ and determining the inflection points.

Steps to Find Inflection Points:

1. Definition: Inflection points occur where the second derivative changes sign. This means we need to analyze where $f''(x) = 0$ or is undefined.

2. Find where $f''(x) = 0$: The second derivative is a product of three terms: $x^2$, $(x - 2)^4$, and $(x - 3)^2$. The only way this product can be zero is if one of these terms is zero: $x^2 = 0 \quad \Rightarrow \quad x = 0$ $(x - 2)^4 = 0 \quad \Rightarrow \quad x = 2$ $(x - 3)^2 = 0 \quad \Rightarrow \quad x = 3$ So, the critical points where $f''(x) = 0$ are $x = 0, 2, 3$.

3. Check for sign changes around each critical point: For each critical point, we analyze the sign of $f''(x)$ around those values to see if the sign changes, which would indicate an inflection point.

   • At $x = 0$: $f''(x) = x^2$ will always be positive for $x \neq 0$. Around $x = 0$, the second derivative remains non-negative (since squaring preserves the sign). Thus, no sign change occurs at $x = 0$, meaning $x = 0$ is not an inflection point.

   • At $x = 2$: The term $(x - 2)^4$ is a fourth power, which is always positive except at $x = 2$, where it is zero. The behavior around $x = 2$ does not lead to a sign change in $f''(x)$, so $x = 2$ is also not an inflection point.

   • At $x = 3$: Similarly, $(x - 3)^2$ is a squared term, and it does not change sign around $x = 3$. Therefore, no inflection point occurs at $x = 3$.

Conclusion:

None of the points $x = 0, 2, 3$ result in a sign change in the second derivative. Therefore, there are no inflection points for this function.

The correct answer is (A) none.

Would you like further details or explanations on any of these steps? Here are 5 related questions for practice:

1. How do you determine the sign of the second derivative for inflection points?
2. Why is it important to check for sign changes in the second derivative?
3. What is the difference between a critical point and an inflection point?
4. Can there be an inflection point if $f''(x) = 0$ but the sign doesn't change?
5. How do the powers of terms (like $x^2$ or $(x - a)^n$) influence the behavior of the function?

Tip: Inflection points occur only when the second derivative changes sign, so always check the behavior of the derivative before and after each critical point.