Question

If f(x) is a differentiable function such that F : R $\to$ R and $f\left(\frac{1}{n}\right)=0\forall n\le 1,nϵI$ then
[IIT Screening 2005]

• A. $f\left(x\right)=0\forall xϵ(0,1)$
• B. f(0) = 0 = f'(0)
• C. f(0) = 0 but f (0) may or may not be 0
• D. $|f\left(x\right)|\le 1\forall xϵ(0,1)$

Answer 1

The correct option is B f(0) = 0 = f'(0)

$f\left(1\right)=f\left(\frac{1}{2}\right)=f(\left(\frac{1}{3}\right)=.....=\underset{n\to \infty }{lim}f(\left(\frac{1}{n}\right)=0$
Since there are infinitely many points in x $ϵ$ (0, 1) where f(x) = 0 and $\underset{n\to \infty }{lim}f\left(\frac{1}{n}\right)=0\Rightarrow f\left(0\right)=0$
And since there are infinitely many points in the neighbourhood of x = 0 such that $\Rightarrow f\left(x\right)$ remains constant in the neighbourhood of x = 0 $\Rightarrow$ f'(0) = 0.