Question

For the function $f\left(x\right)=\left[x\right]+\ln x$, which of the following is true regarding its monotonicity at $x=\frac{7}{2}$
$\left(\right[\cdot ]$ denotes the greatest integer function$)$

• A. Increasing
• B. Decreasing
• C. Non-monotonic
• D. Cannot be determined

Answer 1

The correct option is A Increasing

$f^{\prime }\left(x\right)=0+\frac{1}{x}$
$\{\because [x]\text{is constant in the interval}x\in (3,4\left)\right\}$
$\Rightarrow f^{\prime }\left(\frac{7}{2}\right)=\frac{2}{7}>0$
$\Rightarrow f\left(x\right)$ is an increasing function.