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JEE Main & Advanced Sample Paper JEE Main Sample Paper-1

  • question_answer
    If \[f(x)=\frac{1}{{{x}^{2}}}\int_{4}^{x}{\{(4{{t}^{2}}-2f'(t)\}}\]dt, then f'(4) is equal to

    A) 32                                          

    B)  \[\frac{32}{3}\]

    C)  \[\frac{32}{9}\]                               

    D)  None of these

    Correct Answer: C

    Solution :

    Given, \[f(x)=\frac{1}{{{x}^{2}}}\int_{4}^{x}{(4{{t}^{2}}-2f'(t)\}dt}\] On differentiating both sides, we get \[f'(x)=\frac{1}{{{x}^{2}}}[4{{x}^{2}}-2f'(x)]\] \[-\frac{2}{{{x}^{3}}}\int_{4}^{x}{[4{{t}^{2}}-2f'(t)]dt}\] \[\Rightarrow \]\[f'(4)=\frac{1}{16}[64-2f'(4)]-0\] \[\Rightarrow \]\[f'(4)\left( 1+\frac{1}{8} \right)=4\]\[\Rightarrow \]\[f'(4)=\frac{32}{9}\]


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