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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Maxima and Minima

  • question_answer
    In \[(-4,\,4)\] the function \[f(x)=\int\limits_{-10}^{x}{({{t}^{4}}-4){{e}^{-4t}}dt}\] has [AMU 2002]

    A)             No extrema

    B)             One extremum

    C)             Two extrema

    D)             Four extrema

    Correct Answer: C

    Solution :

               \[f(x)=\int_{-10}^{x}{({{t}^{4}}-4){{e}^{-4t}}dt}\] Þ \[{f}'(x)=({{x}^{4}}-4){{e}^{-4x}}\]            Now  \[{f}'(x)=0\Rightarrow x=\pm \sqrt{2},\,\pm \sqrt{2}\]            Now \[{f}''(x)=-\,4({{x}^{4}}-4){{e}^{-4x}}+4{{x}^{3}}{{e}^{-4x}}\]             At \[x=\sqrt{2}\] and \[x=-\sqrt{2}\] the given function has extreme value.


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