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To determine the inflection points of the function \( f(x) = x^4 - 32x^2 + 16 \), we need to follow these steps:

1. **Find the second derivative of \( f(x) \)**: Inflection points occur where the second derivative changes sign.

### Step 1: First Derivative
Given:
\[
f(x) = x^4 - 32x^2 + 16
\]
The first derivative \( f'(x) \) is:
\[
f'(x) = 4x^3 - 64x
\]

### Step 2: Second Derivative
Now, let's differentiate \( f'(x) \) to get \( f''(x) \):
\[
f''(x) = 12x^2 - 64
\]

### Step 3: Set \( f''(x) = 0 \)
To find potential inflection points, set the second derivative equal to zero and solve for \( x \):
\[
12x^2 - 64 = 0
\]
\[
12x^2 = 64
\]
\[
x^2 = \frac{64}{12} = \frac{16}{3}
\]
\[
x = \pm \sqrt{\frac{16}{3}} = \pm \frac{4}{\sqrt{3}} = \pm \frac{4\sqrt{3}}{3}
\]

### Step 4: Verify the Change in Concavity
To confirm these points are indeed inflection points, we should check the sign of \( f''(x) \) around \( x = \pm \frac{4\sqrt{3}}{3} \). If \( f''(x) \) changes sign around these values, they are inflection points.

Since \( f''(x) \) does indeed change sign around \( x = \pm \frac{4\sqrt{3}}{3} \), these values are inflection points.

### Answer
\[
x = \frac{4\sqrt{3}}{3}, -\frac{4\sqrt{3}}{3}
\]

Would you like a deeper explanation of any step?

Here are some related questions to consider:

1. How do you determine the concavity of a function based on its second derivative?
2. Why is it necessary for the second derivative to change signs for an inflection point?
3. What are some examples of functions without any inflection points?
4. How does the degree of a polynomial function affect the number of possible inflection points?
5. How can we find the intervals where a function is concave up or concave down?

**Tip:** Inflection points are often where the function changes from being "curved up" to "curved down" or vice versa. To verify an inflection point, always check for a sign change in the second derivative around the point.

Determine the exact values at which inflection points occur, if any, of the function f(x) = x^4 - 32x^2 + 16. Give your answer as a comma-separated list. If the function does not have any inflection points, enter DNE.

Learn how to find the exact values of inflection points for the function f(x) = x^4 - 32x^2 + 16. This solution uses the second derivative test and explores concavity changes to identify where inflection points occur. Suitable for advanced high school calculus students.

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