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JEE Main & Advanced Mathematics Differentiation Question Bank Self Evaluation Test - Limits and Derivatives

  • question_answer
    Let \[f(x)=\alpha (x)\beta (x)\gamma (x)\] for all real x, where \[\alpha (x),\beta (x)\] and \[\gamma (x)\] are differentiable functions of\[x.\] If \[f'(2)=18f(2),\alpha '(2)=3\alpha (2),\beta '(2)=-4\beta (2)\] and \[\gamma '(2)-k\gamma (2),\] then the value of k is

    A) 14

    B) 16

    C) 19

    D) None of these  

    Correct Answer: C

    Solution :

    [c] We have, \[f(x)=\alpha (x)\beta (x)\gamma (x),\]for all real x. \[\Rightarrow f'(x)=\alpha '(x)\beta (x)\gamma (x)+\alpha (x)\beta '(x)\gamma (x)+\alpha (x)\beta (x)\gamma '(x)\]\[\Rightarrow f'(2)=\alpha '(2)\beta (2)\gamma (2)+\alpha (2)\beta '(2)\gamma (2)+\alpha (2)\beta (2)\gamma '(2)\]\[\Rightarrow 18f(2)=3\alpha (2)\beta (2)\gamma (2)-4\alpha (2)\beta '(2)\gamma (2)+k\alpha (2)\beta (2)\gamma (2)\]\[[\because f'(2)=18f(2),\alpha '(2)=3\alpha (2),\beta '(2)\] \[=-4\beta (2)and\gamma '(2)=k\gamma (2)]\] \[\Rightarrow 18f(2)=(-1+k)\alpha (2)\beta (2)\gamma (2)=(k-1)f(2)\] \[[\because f(2)=\alpha (2)\beta (2)\gamma (2)]\] \[\Rightarrow k-1=18\Rightarrow k=19.\]


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