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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Self Evaluation Test - Application of Derivatives

  • question_answer
    If \[f(x)=k{{x}^{3}}-9{{x}^{2}}+9x+3\] is monotonically increasing in every interval, then which one of the following is correct?

    A) \[k<3\]

    B) \[k\le 3\]

    C) \[k>3\]

    D) \[k\ge 3\]

    Correct Answer: C

    Solution :

    [c] Given \[f(x)=k{{x}^{3}}-9{{x}^{2}}+9x+3\] On differentiating w.r.t.x, we get \[f'(x)=3k{{x}^{2}}-18x+9\] For a function to be monotonically increasing. \[{{b}^{2}}-4ac<0\] Here, \[a=3k,b=-18,c=9\] \[\therefore {{b}^{2}}-4ac={{(-18)}^{2}}-4(3k)(9)\]                              \[=(-18)(-18)-(3k)18\times 2\] \[\Rightarrow 36-12k<0\Rightarrow k>3\]


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