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JEE Main & Advanced Mathematics Differentiation Question Bank Differentiation by Substitution

  • question_answer
    If \[f(x)\] is a differentiable function and \[{f}''(0)=a\] then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}\] is                                                            [Orissa JEE 2004]

    A)            3a

    B)            2a

    C)            5a

    D)            4a

    Correct Answer: A

    Solution :

               \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f(x)-3f(2x)+f(4x)}{{{x}^{2}}}\]                    Using L-Hospital?s rule twice, we get                    \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2f''(x)-3.2.2f''(2x)+4.4f''(4x)}{2}=3a\] {\[\because \]Given \[f''(0)=a\]}.


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