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JEE Main & Advanced Mathematics Applications of Derivatives Question Bank Increasing and Decreasing Function

  • question_answer
    The interval in which the function \[{{x}^{3}}\]increases less rapidly than\[6{{x}^{2}}+15x+5\], is

    A)            \[(-\infty ,\,-1)\]

    B)            (?5 , 1)

    C)            (?1 ,5)

    D)            (5 , \[\infty \])

    Correct Answer: C

    Solution :

               The function \[f(x)={{x}^{3}}\]increases for all x and the function \[g(x)=6{{x}^{2}}+15x+5\]increases, if                    \[g'(x)>0\Rightarrow 12x+15>0\Rightarrow x>-\frac{5}{4}\].                    Thus \[f(x)\] and \[g(x)\]both increases for \[x>-\frac{5}{4}\].                    It is given that\[f(x)\]increases less rapidly than \[g(x)\],            Therefore the function \[\varphi (x)=f(x)-g(x)\]is decreasing  function , which implies that \[\varphi '(x)<0\]                    Þ \[3{{x}^{2}}-12x-15<0\Rightarrow -1<x<5\].


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