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

  • question_answer
    Let \[f\,(x)=1+2{{x}^{2}}+{{2}^{2}}{{x}^{4}}+.....+{{2}^{10}}{{x}^{20}}\],  then \[f(x)\] has                                                               [AMU 2005]

    A)            More than one minimum

    B)            Exactly one minimum

    C)            At least one maximum

    D)            None of these

    Correct Answer: B

    Solution :

               \[f(x)=1+2{{x}^{2}}+{{2}^{2}}{{x}^{4}}+{{2}^{3}}{{x}^{6}}+{{...2}^{10}}{{x}^{20}}\]                    \[{f}'(x)=x(4+{{4.2}^{2}}{{x}^{2}}+....+{{20.2}^{10}}{{x}^{18}})\]                    \[\therefore \] \[{f}'(x)=0\] Þ \[x=0\] only.                    Also \[{f}''(0)>0\].


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