For all twice differentiable functions f:R→R, with f(0)=f(1)=f′(0)=0,
A
f′′(x)=0, at every point x∈(0,1)
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B
f′′(x)≠0, at every point x∈(0,1)
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C
f′′(x)=0, for some x∈(0,1)
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D
f′′(0)=0
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Solution
The correct option is Cf′′(x)=0, for some x∈(0,1) Applying rolle`s theorem in [0,1] for function f(x) f′(c)=0,c∈(0,1)
again applying rolles theorem in [0,c] for function f′(x) ⇒f′′(c1)=0,c1∈(0,c)