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We are given the second derivative of a function $f$ as:
$f''(x) = x^2 (x - 2)^4 (x - 3)^2$ We need to find the x-coordinates of the inflection points. Inflection points occur where the second derivative changes sign.

Steps to Solve:

1. Identify Critical Points: Set $f''(x) = 0$.
   $x^2 (x - 2)^4 (x - 3)^2 = 0$ This equation is satisfied if any of the following factors is zero:

   • $x^2 = 0 \Rightarrow x = 0$
   • $(x - 2)^4 = 0 \Rightarrow x = 2$
   • $(x - 3)^2 = 0 \Rightarrow x = 3$

   So, the possible inflection points are at $x = 0$, $x = 2$, and $x = 3$.

2. Check Sign Changes around Critical Points: To confirm if these are actual inflection points, we need to see if $f''(x)$ changes sign at these points. We analyze the sign of $f''(x)$ on the intervals determined by $x = 0$, $x = 2$, and $x = 3$.

   • Interval: $(- \infty, 0)$
     Pick $x = -1$:
     $f''(-1) = (-1)^2 \cdot (-1 - 2)^4 \cdot (-1 - 3)^2 > 0$ (positive)

   • Interval: $(0, 2)$
     Pick $x = 1$:
     $f''(1) = 1^2 \cdot (1 - 2)^4 \cdot (1 - 3)^2 > 0$ (positive)

   • Interval: $(2, 3)$
     Pick $x = 2.5$:
     $f''(2.5) = (2.5)^2 \cdot (2.5 - 2)^4 \cdot (2.5 - 3)^2 > 0$ (positive)

   • Interval: $(3, \infty)$
     Pick $x = 4$:
     $f''(4) = 4^2 \cdot (4 - 2)^4 \cdot (4 - 3)^2 > 0$ (positive)

Conclusion:

• There is no sign change around any of the points $x = 0$, $x = 2$, or $x = 3$.
• Therefore, there are no inflection points.

Answer:

The correct option is:
(A) none

Would you like further clarifications or additional practice problems?

Related Questions:

1. How do you find inflection points for functions involving trigonometric terms?
2. What’s the difference between critical points and inflection points?
3. Can inflection points occur where the second derivative is undefined?
4. What happens to the concavity of a function at an inflection point?
5. How does the second derivative test relate to finding maxima or minima?

Tip: When dealing with high powers like $(x - 2)^4$, always check sign changes to confirm inflection points.