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JEE Main & Advanced JEE Main Paper (Held on 08-4-2019 Morning)

  • question_answer
    Let \[f:[0,2]\to R\]be a twice differentiable function such that \[''\left( x \right)>0,\] for all\[x\in (0,2)\]. If \[\phi (x)=f(x)+f(2-x),\] then \[\phi \]is :                   [JEE Main 8-4-2019 Morning]

    A) decreasing on (0, 2)

    B) decreasing on (0, 1) and increasing on (1, 2)

    C) increasing on (0, 2)

    D) increasing on (0, 1) and decreasing on (1, 2)

    Correct Answer: B

    Solution :

    \[\phi (x)=f(x)+f(2-x)\] \[\phi '(x)=f'(x)-f'(2-x)\]       ...... Since \[f''(x)>0\] \[\Rightarrow \]\[f'(x)\] is increasing \[\forall x\in (0,2)\] Case-I : When \[x>2-x\Rightarrow x>1\] \[\Rightarrow \]\[\phi '(x)>0\forall x\in (1,2)\] \[\therefore \]\[\phi (x)\] is increasing on (1, 2) Case-II :When \[x<2-x\Rightarrow x<1\] \[\phi '(x)<0\forall x\in (0,1)\] \[\therefore \]\[\phi (x)\] is decreasing on (0, 1)


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