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Question

Let f:R→R be a function such that |f(x)|≤x2, for all x∈R. At x=0,f is

A
Continuous but not differentiable
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B
Continuous as well as differentiable
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C
Neither continuous nor differentiable
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D
Differentiable but not continuous
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Solution

The correct option is B Continuous as well as differentiableSince, |f(x)|≤x2 ∀ x∈R , we have at x=0,|f(0)|≤0 ∴f(0)=0 ...(1) ∴f′(0)=limh→0f(h)−f(0)h ⇒f′(0)=limh→0f(h)h ...(2) Now, ∣∣∣f(h)h∣∣∣≤|h| (∵|f(x)|≤x2) ⇒−|h|≤f(h)h≤|h| Applying limit on the above equation −limh→0|h|≤limh→0f(h)h≤limh→0|h| By using sandwich theorem, limh→0f(h)h=0 ...(3) Therefore, from eqn(2) and (3), we get f′(0)=0 i.e., f(x) is differentiable at x=0.

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