Let f(x) be a function given by $f(x)=\frac{x^2+1}{[x]}$ for all $x \in[1,4)$, where [.] denotes the greatest integer function. Then, f(x) is monotonically

increasing on $[1,2)$

We have, $f(x)=\frac{x^2+1}{[x]}$ for all $x \in[1,4)$

$\Rightarrow f(x)= \begin{cases} {x^2+1}, & \text { for all } x \in[1,2) \\ \frac{x^2+1}{2}, & \text { for all } x \in[2,3) \\ \frac{x^2+1}{3}, & \text { for all } x \in[3,4)\end{cases}$

$\Rightarrow f^{\prime}(x)=\left\{\begin{aligned} 2 x, & \text { for all } x \in[1,2) \\ x, & \text { for all } x \in[2,3) \\ \frac{2 x}{3}, & \text { for all } x \in(3,4)\end{aligned}\right.$

$f(x)$ values in (1, 2) are greater than $f(x)$ values in (2, 3) here $f(x)$ is increasing only on (1, 2) not (1, 4).