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J & K CET Engineering J and K - CET Engineering Solved Paper-2008

  • question_answer
    The function \[f(x)={{x}^{2}}{{e}^{-x}}\] increases in the interval

    A)  \[(0,2)\]         

    B)  \[(2,3)\]

    C)  \[(3,4)\]

    D)  \[(4,5)\]

    Correct Answer: A

    Solution :

    Given, \[f(x)={{x}^{2}}{{e}^{-x}}\] \[\Rightarrow \] \[f'(x)=2x{{e}^{-x}}-{{x}^{2}}{{e}^{-x}}\] For \[f(x)\] to be increasing, \[f'(x)>0\] \[\Rightarrow \] \[2x{{e}^{-x}}-{{x}^{2}}{{e}^{-x}}>0\] \[\Rightarrow \] \[{{e}^{-x}}(2x-{{x}^{2}})>0\] \[\Rightarrow \] \[x(2-x)>0\] \[\Rightarrow \] \[x\in (0,2)\]


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