Question

If f(x) is a continuous and differentiable function and $f\left(\frac{1}{n}\right)=0\forall n\ge 1$ and $n\in I$, then

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

Answer 1

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

$f\left(\frac{1}{n}\right)=0n\ge 1,n\in I$

The function is equal to $0$ only when n is integer and so f is not zero for other values like $\frac{1}{1.5}$ etc.,

The function has wavy nature since it is zero for many n (integers)

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

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

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

$\Rightarrow f^{\prime }\left(\frac{1}{n}\right)\left(\frac{-1}{n^2}\right)=0$

$\Rightarrow f^{\prime }\left(\frac{1}{n}\right)=0$

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

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