A) \[f(x)=0\ \forall \ x\in (0,\,1)\]
B) \[f(x)=0\forall x\in (0,\,1)\]
C) \[f(0)=0\] but \[f'(0)\] may or may not be 0
D) \[|f(x)|\,\le 1\ \forall \ x\in (0,\,1)\]
Correct Answer: B
Solution :
\[f(1)=f\left( \frac{1}{2} \right)=f\left( \frac{1}{3} \right)=......=\underset{n\to \infty }{\mathop{\lim }}\,f\left( \frac{1}{n} \right)=0\] Since there are infinitely many points in \[x\in (0,\,1)\] where \[f(x)=0\] and \[\underset{n\to \infty }{\mathop{\lim }}\,f\left( \frac{1}{n} \right)=0\] 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 \[x=0\] Þ \[f'(0)=0\].
You need to login to perform this action.
You will be redirected in
3 sec
