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

  • question_answer
    Maximum slope of the curve \[y=-{{x}^{3}}+3{{x}^{2}}+9x-27\] is                                                                           [MP PET 2001]

    A)            0

    B)            12

    C)            16

    D)            32

    Correct Answer: B

    Solution :

               \[y=f(x)=-{{x}^{3}}+3{{x}^{2}}+9x-27\]            The slope of this curve \[{f}'(x)=-3{{x}^{2}}+6x+9\]            Let \[g(x)={f}'(x)=-3{{x}^{2}}+6x+9\]            Differentiate with respect to x, \[{g}'(x)=-6x+6\]            Put \[{g}'(x)=0\] Þ \[x=1\]            Now, \[{g}''(x)=-6<0\] and hence at \[x=1,\] \[g(x)\] (slope) will have maximum value.            \ \[{{[g(1)]}_{\text{max}\text{.}}}=-3\times 1+6+9=12\].


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