To find the critical numbers of the function \( f(x) = \frac{4x}{x^2 + 1} \), we first need to find its derivative and determine where it is equal to zero or undefined. These points are the critical numbers.

### Step 1: Find the derivative of \( f(x) \)
We'll use the **quotient rule** to differentiate. The quotient rule states:

\[
\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}
\]

For \( f(x) = \frac{4x}{x^2 + 1} \), let:
- \( u = 4x \), so \( u' = 4 \),
- \( v = x^2 + 1 \), so \( v' = 2x \).

Now, apply the quotient rule:

\[
f'(x) = \frac{(x^2 + 1)(4) - (4x)(2x)}{(x^2 + 1)^2}
\]

Simplify the numerator:

\[
f'(x) = \frac{4(x^2 + 1) - 8x^2}{(x^2 + 1)^2}
\]

\[
f'(x) = \frac{4x^2 + 4 - 8x^2}{(x^2 + 1)^2}
\]

\[
f'(x) = \frac{-4x^2 + 4}{(x^2 + 1)^2}
\]

\[
f'(x) = \frac{4(1 - x^2)}{(x^2 + 1)^2}
\]

### Step 2: Set the derivative equal to zero
To find the critical numbers, we need to solve \( f'(x) = 0 \):

\[
\frac{4(1 - x^2)}{(x^2 + 1)^2} = 0
\]

The fraction equals zero when the numerator is zero, so set the numerator \( 4(1 - x^2) = 0 \):

\[
1 - x^2 = 0
\]

\[
x^2 = 1
\]

\[
x = \pm 1
\]

### Step 3: Check for undefined points
Since the denominator \( (x^2 + 1)^2 \) is never zero for any real \( x \), the derivative is never undefined.

### Conclusion
The critical numbers of \( f(x) = \frac{4x}{x^2 + 1} \) are \( x = 1 \) and \( x = -1 \).

Would you like more details on finding critical numbers or help with graphing this function? 

Here are 5 related questions:
1. How do we classify critical points as maxima, minima, or points of inflection?
2. What is the behavior of the function \( f(x) = \frac{4x}{x^2 + 1} \) near \( x = 1 \) and \( x = -1 \)?
3. How do we analyze the concavity of a function using the second derivative?
4. How do we find horizontal asymptotes of rational functions?
5. How do we compute the global maximum or minimum of \( f(x) = \frac{4x}{x^2 + 1} \)?

**Tip**: When solving optimization problems, always verify critical points using the first or second derivative test to confirm if they are maxima or minima.

This page provides a step-by-step guide to finding the critical numbers of the function f(x) = 4x/(x^2 + 1). Learn how to apply the quotient rule to differentiate and solve for critical points. Ideal for high school and college students studying calculus.

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