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

  • question_answer
    If sum of two numbers is 3, then maximum value of the product of first and the square of second is [MP PET 1996]

    A)            4

    B)            3

    C)            2

    D)            1

    Correct Answer: A

    Solution :

               Let the first number be \[3-x\], then second will be \[x\].            Accordingly, we have to maximize \[(3-x){{x}^{2}}\]            Let \[f(x)=(3-x){{x}^{2}}=3{{x}^{2}}-{{x}^{3}}\Rightarrow f'(x)=6x-3{{x}^{2}}\]            \[\therefore f'(x)=0\Rightarrow x=0,\ 2\]            Also \[{f}''\,(x)=6-6x\]. Obviously, \[{f}''\,(2)=-6<0\].            Therefore, required max. value \[=(3-2).\,{{2}^{2}}=4.\]


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