The largest open interval, in which the function $f(x)=\frac{x}{x^2+1}$ increases, is
Answer & explanation
Correct answer: option 3
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:
$1 - x^2 > 0 \Rightarrow x^2 < 1 \Rightarrow -1 < x < 1$