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Question

Let f:RR be twice continuously differentiable (or f′′ exists and is continuous) such that f(0)=f(1)=f(0)=0. Then

A
f′′(c)=0 for some cR
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B
there is no point for which f′′(x)=0
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C
at all points f′′(x)>0
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D
at all points f′′(x)<0
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Solution

The correct option is A f′′(c)=0 for some cR
Consider f(x) on [0,1]
Applying Rolle's theorem on the interval [0,1],
f(a)=0 for some a(0,1)

Now, applying Rolle's theorem to f(x)on the interval [0,a],
f′′(c)=0 for some c(0,a)

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