# If $f(x)$ is continuous and differentiable function and $f(1/n)=0\forall n \geq 1$ and $n\in 1$ then

$\begin{array}{1 1}(a)\;f(x)=0\;x\in (0,1)\\(b)\;f(0)=0,f'(0)=0\\(c)\;f(0)=0=f'(0)\;x\in(0,1)\\(d)\;f(0)=0\;and\;f'(0)\;need\;not\;be\;zero\end{array}$

Given that $f(x)$ is a continuous and differentiable function and $f(\large\frac{1}{x})$$=0,x=n,n\in I \therefore f(0^+)=f(\large\frac{1}{\infty})=$$0$
Since RHL=0
$f(0)=0$ for $f(x)$ to be continuous.
Also $f'(0)=\lim\limits_{h\to 0}\large\frac{f(h)-f(0)}{h-0}$
$\Rightarrow \lim\limits_{h\to 0}\large\frac{f(h)}{h}$$=0$
Using f(0)=0 and $f(0^+)=0$
Hence $f(0)=0$ and $f'(0)=0$
Hence (b) is the correct answer.