Question

$f\left(x\right)=1+\left[\cos x\right]$, in $0<x\le \frac{\pi }{2}$
(where $[.]$ denotes greatest integer function)

• A. has a minimum value of $0$
• B. has a maximum value of $2$
• C. is continuous in $[0,\frac{\pi }{2}]$
• D. is not differentiable at $x=\frac{\pi }{2}$

Answer 1

The correct option is C is continuous in $[0,\frac{\pi }{2}]$

Given $f\left(x\right)=1+\left[\cos x\right]$, in $0<x\le \frac{\pi }{2}$
Since, $\left[\cos x\right]=0$ for $0<x\le \frac{\pi }{2}$
So, $f\left(x\right)=1$ for $0<x\le \frac{\pi }{2}$
which is a constant function.
$\therefore$ $f\left(x\right)$ is continuous in $[0,\frac{\pi }{2}]$
Since, f(x) is constant function and constant functions are differentiable in its entire domain.
Hence, it is differentiable at $x=\frac{\pi }{2}$