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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Self Evaluation Test - Application of Derivatives

  • question_answer
    Let \[g(x)=2f\left( \frac{x}{2} \right)+f(2-x)\] and \[f''(x)<0\forall x\in (0,2)\]. Then g(x) increases in

    A) (1/2, 2)

    B) (4/3, 2)

    C) (0, 2)

    D) (0, 4/3)

    Correct Answer: D

    Solution :

    [d] We have \[g'(x)=f'\left( \frac{x}{2} \right)-f'(2-x)\] Given \[f''(x)<0\forall \in (0,2)\] So, \[f'(x)\] is decreasing on (0, 2). Let \[\frac{x}{2}>2-x\] or \[f'\left( \frac{x}{2} \right)<f'(2-x)\]. Thus, \[\forall x>\frac{4}{3},g'(x)<0\]. Therefore, \[g(x)\] decreasing in \[\left( \frac{4}{3},2 \right)\] and increasing in \[\left( 0,\frac{4}{3} \right)\].


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