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To find the relative extrema for the function \( f(x) = x^4 + 32x^2 \), we need to follow these steps:

### Step 1: Find the first derivative of \( f(x) \)
We differentiate \( f(x) = x^4 + 32x^2 \) with respect to \( x \).

\[
f'(x) = 4x^3 + 64x
\]

### Step 2: Set the first derivative equal to zero and solve for \( x \)
To find critical points, we set \( f'(x) = 0 \):

\[
4x^3 + 64x = 0
\]

Factoring out \( 4x \):

\[
4x(x^2 + 16) = 0
\]

Thus, the equation becomes:

\[
x = 0 \quad \text{or} \quad x^2 = -16
\]

Since \( x^2 = -16 \) has no real solutions, the only real critical point is \( x = 0 \).

### Step 3: Determine whether the critical point is a maximum or minimum using the second derivative test
We now find the second derivative of \( f(x) \):

\[
f''(x) = 12x^2 + 64
\]

Substituting \( x = 0 \) into the second derivative:

\[
f''(0) = 12(0)^2 + 64 = 64
\]

Since \( f''(0) = 64 > 0 \), the function is concave up at \( x = 0 \), meaning there is a **relative minimum** at \( x = 0 \).

### Step 4: Find the corresponding value of \( f(x) \)
To find the value of the function at \( x = 0 \):

\[
f(0) = (0)^4 + 32(0)^2 = 0
\]

Thus, the relative minimum occurs at the point \( (0, 0) \).

### Conclusion:
- **Relative maximum**: DNE (there are no relative maxima).
- **Relative minimum**: \( (0, 0) \).

Would you like more details or have any questions?

Here are five related questions:
1. What are the conditions for using the second derivative test?
2. How would the result change if the second derivative was negative at \( x = 0 \)?
3. Can you explain how factoring helps in solving \( f'(x) = 0 \)?
4. What is the meaning of a "relative extremum" in terms of a graph?
5. How can you confirm if a point is an inflection point?

**Tip:** The second derivative test helps you determine concavity, which is key in identifying relative extrema.

Find the relative extrema for f(x) = x^4 + 32x^2. Enter your answers as an ordered pair.

Learn how to find the relative extrema for the function f(x) = x^4 + 32x^2 by applying derivatives. This solution involves calculating the first and second derivatives and using the second derivative test to determine concavity and identify relative minima or maxima.

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