Functions Question 406
If $\mathrm{f}(\mathrm{x})$ is a differentiable function such that $\mathrm{f} : R \rightarrow R$ and $f\left(\frac{1}{n}\right)=0 \forall n \geq 1, n \in \mathbb{N}$, then
[IIT Screening 2005]
Options:
A) $ f(x)=0\ \forall \ x\in (0, 1) $
B) $f(0)=0=f^{\prime}(0)$
C) $ f(0)=0 $ but $ f’(0) $ may or may not be 0
D) $ |f(x)| \le 1\ \forall \ x\in (0, 1) $
Show Answer
Answer:
Correct Answer: B
Solution:
$ f(1)=f( \frac{1}{2} )=f( \frac{1}{3} )=……=\underset{n\to \infty }{\mathop{\lim }} f( \frac{1}{n} )=0 $ Since there are infinitely many points in $ x\in (0, 1) $ where $ f(x)=0 $ and $ \underset{n\to \infty }{\mathop{\lim }} f( \frac{1}{n} )=0 $
Þ $ f(0)=0 $ And since there are infinitely many points in the neighbourhood of $ x=0 $ such that
Þ $ f(x) $ remains constant in the neighbourhood of $ x=0 $ is not necessarily true; it depends on the specific function $ f(x) $.
Þ $ f’(0)=0 $ .
BETA
