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J & K CET Engineering J and K - CET Engineering Solved Paper-2015

  • question_answer
    In which of the following interval, the function \[y(x)={{x}^{3}}-3{{x}^{2}}-9x+5\] is always decreasing?

    A)  \[(-1,3)\]              

    B)  \[(-3,3)\]

    C)  \[(-4,4)\]              

    D)  \[(-2,2)\]

    Correct Answer: A

    Solution :

    Given,  \[f(x)={{x}^{3}}-3{{x}^{2}}-9x+5\] \[f'(x)=3{{x}^{2}}-6x-9\] For maxima or minima, \[f'(x)=0\] \[\Rightarrow \] \[3{{x}^{2}}-6x-9=0\] \[\Rightarrow \] \[3({{x}^{2}}-2x-3)=0\] \[\Rightarrow \] \[{{x}^{2}}-2x-3=0\] \[\Rightarrow \] \[{{x}^{2}}-3x+x-3=0\] \[\Rightarrow \] \[x(x-3)+1(x-3)=0\] \[\Rightarrow \] \[(x+1)(x-3)=0\] \[\Rightarrow \] \[x=-1,3\] \[\therefore \]  Intervals are \[(-\infty ,-1),\,(-1,3)\] and \[(3,\,\infty )\].
    Intervals Sign of \[f'(x)\] Nature of \[f'(x)\]
    \[(-\infty ,-1)\] \[+ve\] Strictly increasing
    \[(-1,3)\] \[-ve\] Strictly decreasing
    \[(3,\infty )\] \[+ve\] Strictly increasing
    So, \[f(x)\] is strictly decreasing in interval \[(-1,3).\]


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