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JEE Main & Advanced Mathematics Limits, Continuity and Differentiability Question Bank Self Evaluation Test - Continuity and Differentiability

  • question_answer
    Let f be a function which is continuous and differentiable for all real x. If \[f(2)=-4\] and \[f'(x)\ge 6\] for all \[x\in [2,4],\] then

    A) \[f(4)<8\]

    B) \[f(4)\ge 8\]

    C) \[f(4)\ge 12\]

    D) None of these

    Correct Answer: B

    Solution :

    [b] By mean value theorem, there exists a real number \[c\in (2,4)\] such that \[f'(c)=\frac{f(4)-f(2)}{4-2}\Rightarrow f'(c)=\frac{f(4)+4}{2}\] Since, \[f'(c)\ge 6,\forall x\in [2,4]\] \[\therefore \,\,\,\,f'(c)\ge 6,\Rightarrow \frac{f(4)+4}{2}\ge 6\] \[\Rightarrow f(4)+4\ge 12\Rightarrow f(4)\ge 8\].


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