Question

The function $f\left(x\right)=1+x\left(\sin \left(x\right)\right)\left[\cos \left(x\right)\right]$, $0<x\le \frac{\pi }{2}$

• A. is continuous on $\left(0,\frac{\pi }{2}\right)$
• B. is strictly increasing on $\left(0,\frac{\pi }{2}\right)$
• C. is strictly decreasing on $\left(0,\frac{\pi }{2}\right)$
• D. None of these

Answer 1

The correct option is A

is continuous on $\left(0,\frac{\pi }{2}\right)$

Explanation for the correct option

Given function, $f\left(x\right)=1+x\left(\sin x\right)\left[\cos x\right]$

Since, $0<x\le \frac{\pi }{2}$

$\Rightarrow 0\le \cos x<1$

So, $[\cos x]=0$

$\therefore f\left(x\right)=1$

So, $f\left(x\right)$ is a constant function and is neither an increasing nor decreasing function.

Thus, $f\left(x\right)$ is a continuous function.

Hence, correct option is $A$