Question

If $f\left(x\right)$ is continuous and differentiable function and $f\left(\frac{1}{n}\right)=0$ for all $n\ge 1$ and $nϵI$ then

• A. $f\left(x\right)=0,xϵ(0,1]$
• B. $f\left(x\right)=0,f^{\prime }\left(0\right)=0$
• C. $f\left(x\right)=0=f^{\prime \prime }\left(0\right),xϵ(0,1]$
• D. $f\left(0\right)=0$ and $f^{\prime }\left(0\right)$ need not zero

Answer 1

The correct option is B $f\left(x\right)=0,f^{\prime }\left(0\right)=0$

Given $f\left(1\right)=f\left(\frac{1}{2}\right)=f\left(\frac{1}{3}\right)=...=\underset{n\to \infty }{lim}f\left(\frac{1}{n}\right)=0$

As $f\left(\frac{1}{n}\right)=0$

$\Rightarrow \underset{n\to \infty }{lim}f\left(\frac{1}{n}\right)=0\Rightarrow f\left(0\right)=0$

Since there are infinitely many points in neighbourhood of $x=0$

$\therefore f\left(x\right)=0\Rightarrow f^{\prime }\left(x\right)=0\Rightarrow f^{\prime }\left(0\right)=0$

Hence $f\left(0\right)=f^{\prime }\left(0\right)=0$