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We are given the function \( f(x) = \frac{x}{x^2 + 1} \), and we need to determine the largest possible value of \( b - a \) where the function is increasing on the interval \( ]a, b[ \).

### Step 1: Find the derivative of the function \( f(x) \)
The derivative of a function tells us whether the function is increasing or decreasing in a given interval. So, we will calculate the derivative of \( f(x) = \frac{x}{x^2 + 1} \) using the quotient rule.

The quotient rule states that for \( f(x) = \frac{u(x)}{v(x)} \), the derivative is given by:
\[
f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
Here, \( u(x) = x \) and \( v(x) = x^2 + 1 \), so:
- \( u'(x) = 1 \)
- \( v'(x) = 2x \)

Now applying the quotient rule:
\[
f'(x) = \frac{(1)(x^2 + 1) - (x)(2x)}{(x^2 + 1)^2}
\]
Simplifying the numerator:
\[
f'(x) = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{-x^2 + 1}{(x^2 + 1)^2}
\]
\[
f'(x) = \frac{1 - x^2}{(x^2 + 1)^2}
\]

### Step 2: Analyze where \( f'(x) \) is positive
For the function to be increasing, the derivative must be positive, i.e., \( f'(x) > 0 \). Therefore, we need:
\[
\frac{1 - x^2}{(x^2 + 1)^2} > 0
\]
The denominator \( (x^2 + 1)^2 \) is always positive, so the sign of \( f'(x) \) depends on the numerator, \( 1 - x^2 \).

Thus, the inequality becomes:
\[
1 - x^2 > 0
\]
\[
x^2 < 1
\]
This implies:
\[
-1 < x < 1
\]

### Step 3: Conclusion
The function is increasing on the interval \( ]-1, 1[ \). Therefore, \( a = -1 \) and \( b = 1 \), so:
\[
b - a = 1 - (-1) = 2
\]

Thus, the greatest value of \( b - a \) is \( \boxed{2} \), which corresponds to option (c).

---

Let me know if you need further details or explanations!

Here are 5 related questions for deeper understanding:
1. How does the quotient rule work for derivatives of rational functions?
2. Why is the denominator of \( f'(x) \) always positive in this case?
3. Can we have an interval where the function is decreasing for this specific function?
4. What happens to \( f(x) \) as \( x \to \infty \)?
5. How can we graphically interpret the interval where the function is increasing?

**Tip:** Always check the sign of the derivative to determine if a function is increasing or decreasing.

If the function f(x) = x / (x^2 + 1) is increasing on ]a,b[, then the greatest of the expression (b-a) = ?

Learn how to find the largest interval where the function f(x) = x / (x^2 + 1) is increasing. This problem covers key calculus concepts such as the quotient rule and derivative analysis. Suitable for Grade 11-12 students studying calculus.

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