To find the critical numbers of the function \( f(x) = \frac{4x}{x^2 + 1} \), we need to:

1. **Find the derivative** of \( f(x) \).
2. **Set the derivative equal to zero** and solve for \( x \).
3. Identify where the derivative does not exist, if applicable, since those points may also yield critical numbers.

### Step 1: Differentiate \( f(x) \)
Using the quotient rule, where if \( f(x) = \frac{g(x)}{h(x)} \), then \( f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \), we can differentiate \( f(x) = \frac{4x}{x^2 + 1} \).

1. Let \( g(x) = 4x \), so \( g'(x) = 4 \).
2. Let \( h(x) = x^2 + 1 \), so \( h'(x) = 2x \).

Using the quotient rule:
\[
f'(x) = \frac{(4)(x^2 + 1) - (4x)(2x)}{(x^2 + 1)^2}
\]
Simplify the numerator:
\[
f'(x) = \frac{4x^2 + 4 - 8x^2}{(x^2 + 1)^2} = \frac{-4x^2 + 4}{(x^2 + 1)^2}
\]
Factor out a \( -4 \) from the numerator:
\[
f'(x) = \frac{-4(x^2 - 1)}{(x^2 + 1)^2} = \frac{-4(x + 1)(x - 1)}{(x^2 + 1)^2}
\]

### Step 2: Set the derivative equal to zero
To find the critical points, set \( f'(x) = 0 \):
\[
\frac{-4(x + 1)(x - 1)}{(x^2 + 1)^2} = 0
\]
The fraction is zero when the numerator is zero, so:
\[
-4(x + 1)(x - 1) = 0
\]
This gives \( x = 1 \) and \( x = -1 \) as solutions.

### Step 3: Check where \( f'(x) \) is undefined
The derivative \( f'(x) \) is undefined when \( (x^2 + 1)^2 = 0 \). However, \( x^2 + 1 = 0 \) has no real solutions, so \( f'(x) \) is defined for all real \( x \).

### Conclusion
The critical numbers of \( f(x) = \frac{4x}{x^2 + 1} \) are:
\[
x = 1 \quad \text{and} \quad x = -1
\]

Would you like further details or have any questions?

Here are 5 additional questions to consider:
1. How would you find the relative extrema at the critical points?
2. What is the behavior of \( f(x) \) as \( x \to \infty \)?
3. Can we analyze concavity to determine points of inflection for \( f(x) \)?
4. How does the behavior of \( f(x) \) compare near \( x = 0 \)?
5. What role does the denominator \( x^2 + 1 \) play in determining the domain of \( f(x) \)?

**Tip:** When working with derivatives of fractions, the quotient rule is essential for simplifying expressions accurately.

Learn how to find the critical numbers of the function f(x) = 4x/(x^2 + 1) using the quotient rule and calculus techniques. This solution covers differentiation, solving for critical points, and identifying undefined derivatives. Suitable for high school students in Grades 11-12.

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