Identify the correct statement where [.] & . denotes greatest int… - Vidyadip

MCQ

Identify the correct statement

where $[.]$ & $\{.\}$ denotes greatest integer function and fractional part function respectively.

A
If $f(x)$ is a differentiable and increasing function then $g(x)$ = $f(f(x)) + 1$ is a decreasing function
B
If $x \in \left( {0,1} \right)$, then $\left[ x \right]\left[ {\sin x} \right] \ne \left[ {x\sin x} \right]$
C
$f\left( x \right) = \left\{ {\cos x} \right\}\left\{ {{{\cos }^2}x} \right\}\left\{ {{{\cos }^3}x} \right\}$ is a continuous function in $\left[ {0,\frac{\pi }{2}} \right]$
✓
$f\left( x \right) = \left\{ x \right\}\left\{ {\sin x} \right\} + \left\{ {x\sin x} \right\}$ is a differentiable function in $x \in \left( {0,1} \right)$

✓

Answer

Correct option: D.

$f\left( x \right) = \left\{ x \right\}\left\{ {\sin x} \right\} + \left\{ {x\sin x} \right\}$ is a differentiable function in $x \in \left( {0,1} \right)$

d
If $x \in(0,1) \quad\{x\}=x$

$\{\sin x\}=\sin x $ and $\{x \sin x\}=x \sin x$

$\therefore f(\mathrm{x})=2 \sin \mathrm{x}$

$\therefore$ differentiable in $\mathrm{x} \in(0,1)$

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