Question

If $f\left(x\right)=\frac{\sin \pi \left[{(5x-7\pi )}^3\right]}{1+{[5x-7\pi ]}^2}$ ($\left[\right]$ denotes the greatest integer function) then

• A. $f\left(x\right)$ is continuous and differentiable in $R$
• B. $f\left(x\right)$ is continuous but not differentiable in $R$
• C. $f^{\prime }\left(x\right)$ does not exits for some values of $x\in R$
• D. None of the above

Answer 1

Answer: (A)

$f\left(x\right)$ is continuous and differentiable in $R$

$f\left(x\right)=\frac{\sin \pi \left[{(5x-7\pi )}^3\right]}{1+{\left[(5x-7\pi )\right]}^2}$ (where $\left[\right]$ deontes GI function)
as ${(5x-7\pi )}^3=$ an integer (say $n$)
$\sin n\pi =0$
and $1+{\left[(5x-7\pi )\right]}^2$ is always non zero number
$\therefore f\left(x\right)=0$
so $f\left(x\right)$ is continuous and differentiable $\forall x\in R$