The largest open interval, in which the function $f(x)=\frac{x}{x^2+1}$ increases, is

$(-1, 1)$

The correct answer is Option (3) → $(-1, 1)$

Given: $f(x) = \frac{x}{x^2 + 1}$

Differentiate: $f'(x) = \frac{(x^2 + 1)(1) - x(2x)}{(x^2 + 1)^2} = \frac{x^2 + 1 - 2x^2}{(x^2 + 1)^2} = \frac{1 - x^2}{(x^2 + 1)^2}$

Now, $f'(x) > 0$ for increasing interval:

$\frac{1 - x^2}{(x^2 + 1)^2} > 0$

Denominator is always positive. So inequality depends on numerator:

$1 - x^2 > 0 \Rightarrow x^2 < 1 \Rightarrow -1 < x < 1$