Question:
Let $f(1,3) \rightarrow R$ be a function defined by $f(x)=\frac{x[x]}{1+x^{2}}$,
where $[x]$ denotes the greatest integer $\leq x$. Then the range of $f$ is:
Correct Option: , 2
Solution:
$f(x) \begin{cases}\frac{x}{x^{2}+1} ; & x \in(1,2) \\ \frac{2 x}{x^{2}+1} ; & x \in[2,3)\end{cases}$
$f^{\prime}(x) \begin{cases}\frac{1-x^{2}}{1+x^{2}} ; & x \in(1,2) \\ \frac{1-2 x^{2}}{1+x^{2}} ; & x \in[2,3)\end{cases}$
$\therefore f(x)$ is a decreasing function
$\therefore \quad y \in\left(\frac{2}{5}, \frac{1}{2}\right) \cup\left(\frac{6}{10}, \frac{4}{5}\right]$
$\Rightarrow \quad y \in\left(\frac{2}{5}, \frac{1}{2}\right) \cup\left(\frac{3}{5}, \frac{4}{5}\right]$