Question

If $f\left(x\right)=1+\left[\cos x\right]x$ in $0<x\le \frac{\pi }{2}$, where $[.]$ denotes the greatest integer function, then which of the following is correct regarding $f\left(x\right)$:

• A. It is continuous and differentiable in $0<x<\frac{\pi }{2}$
• B. $f\left(0\right)=0$
• C. It is not differentiable in $0<x<\frac{\pi }{2}$
• D. It is not differentiable at $x=1$

Answer 1

The correct option is A It is continuous and differentiable in $0<x<\frac{\pi }{2}$

$f\left(x\right)=1+\left[\cos x\right]x$ in $0<x\le \frac{\pi }{2}$
in $0<x\le \frac{\pi }{2},\cos x\in [0,1)$
$\Rightarrow f\left(x\right)=1+0\cdot x=1$
Since, $f\left(x\right)$ is a constant function , it is continuous as well as differentiable in the entire interval.
Clearly, $f\left(0\right)=1$.
Also, $x=1$ lies in the given interval $0<x\le \frac{\pi }{2}$
Hence, $f\left(x\right)$ is differentiable at $x=1$