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  • A function  <img alt="f: \mathbf{R} \rightarrow \mathbf{R}" src="https://entranc

A function  f: \mathbf{R} \rightarrow \mathbf{R} is differentiable and satisfies the condition f\left(\frac{1}{n}\right)=ofor all integers n\geq 1 Then

 

Option: 1

f(x)=0 \text { for all } x \in(0,1]


Option: 2

f(0)=0=f^{\prime}(0)


Option: 3

\mathrm{f}(0)=0 \text { but } f^{\prime}(0) \text { need not be equal to } 0 .


Option: 4

|f(x)| \leq 1 \text { for all } \mathrm{x} \in[0,1]


Answers (1)

best_answer

x_n=\frac{1}{n}(n \in N) \rightarrow 0 \text { as } n \rightarrow \infty

By the definition of continuityf\left(x_n\right) \rightarrow f(0) \text { as } n \rightarrow \infty . But f\left(x_n\right)=0 \text { for all } x \in N \text {. So } \mathrm{f}(0)=0 \text {. }

Also \underset{x \rightarrow 0}{\operatorname{Lt}} \frac{f(x)-f(0)}{x-0}=\underset{x \rightarrow 0}{\operatorname{Lt}} \frac{f\left(x_n\right)-f(0)}{x_n-0} 

 (f is derivable at x = 0)

 

Posted by

Shailly goel

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