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Question

# Let f:R→R be twice continuously differentiable. Let f(0)=f(1)=f′(0)=0. Then

A
f"(x)0 for all x
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B
f"(c)=0 for some cϵR
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C
f"(x)0 if x0
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D
f(x)>0 for all x
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Solution

## The correct option is D f"(c)=0 for some cϵRGiven : f:R→R is twice continuously differentiableAlso, f(0)=f(1)=0By Rolle's theorem, there exist some x between 0 and 1 such that f′(x)=0Now, f′ is also continuously differentiable and f′(0)=0=f′(x) Again applying Rolle's theorem we get f"(c)=0 for some cϵ[0,1].Thus, f"(c)=0 for some cϵR

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