A) \[f(x)=2,\,\,\forall \,\,x\in R\]
B) \[f(x)=5=f(x)\]. For some\[x\in (1,\,\,3)\]
C) there exists atleast one\[x\in (1,\,\,3)\]such that\[f(x)=2\]
D) None of the above
Correct Answer: C
Solution :
Let\[g(x)=f(x)-{{x}^{2}}\] \[\Rightarrow \]\[g(x)\]has atleast 3 real roots which are\[x=1,\,\,2,\,\,3\](by mean value theorem) \[\Rightarrow \]\[g(x)\]has atleast 2 real roots in\[x\in \left( 1,\,\,3 \right)\] \[\Rightarrow \]\[g\,\,(x)\]has atleast 1 real root in\[x\in \left( 1,\,\,3 \right)\] \[f\,\,(x)=2\]for atleast one\[x\in \left( 1,\,\,3 \right)\]
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