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Let \mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} satisfying |\mathrm{f}(\mathrm{x})| \leq \mathrm{x}^2 \forall \mathrm{x} \in \mathrm{R}, then

Option: 1

f is continuous but non-differentiable at x=0


Option: 2

f is discontinuous at x=0


Option: 3

f is differentiable at x=0


Option: 4

none of these.


Answers (1)

best_answer

\text{Putting}\ x=0, \text{we get,}\ |f(x)| \leq 0\\ \begin{aligned} & \Rightarrow f(0)=0 \\ & f^{\prime}(0)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=\lim _{h \rightarrow 0} \frac{f(h)}{h} \\ & \text { Now }\left|\frac{f(h)}{h}\right| \leq|h| \\ & \Rightarrow-|h| \leq \frac{f(h)}{h} \leq|h| \Rightarrow \lim _{h \rightarrow 0} \frac{f(h)}{h}=0 \\ & \Rightarrow f(x) \text { is differentiable at } x=0 \end{aligned}

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Divya Prakash Singh

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