A) \[f''(x)=2,\forall x\in R\]
B) There exist at least one \[x\in (1,\,3)\] such that \[f''(x)=2\]
C) There exist at least one \[x\in (2,\,3)\] such that \[f'(x)=5=f''(x)\]
D) There exist at least one \[x\in (1,\,2)\] such that \[f(x)=3\]
Correct Answer: B
Solution :
Let a function be \[g(x)=f(x)-{{x}^{2}}\] Þ \[g(x)\] has at least 3 real roots which are \[x=1\], 2 , 3 Þ \[g'(x)\] has at least 2 real roots in \[x\in (1,\,3)\] Þ \[g''(x)\]has at least 1 real roots in \[x\in (1,\,3)\] Þ \[f'(x)=2\] for at least one \[x\in (1,3)\].
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