A) Exactly once in (a, b)
B) At most once in (a, b)
C) At least once in (a, b)
D) None of these
Correct Answer: B
Solution :
[b] Suppose, there are two points \[{{x}_{1}}\] and \[{{x}_{2}}\] in (a, b) such that \[f'({{x}_{1}})=f'({{x}_{2}})=0\]. By Rolle?s theorem applied to f? on \[[{{x}_{1}},{{x}_{2}}]\], there must be a \[c\in ({{x}_{1}},{{x}_{2}})\] such that \[f''(c)=0\]. This contradicts the given condition \[f''(x)<0,\forall x\in (a,b)\]. Hence, our assumption is wrong. Therefore, there can be at most one point in (a, b) at which f?(x) is zero.
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