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JEE Main & Advanced JEE Main Paper (Held On 22 April 2013)

  • question_answer
    Statement 1: The function \[{{x}^{2}}({{\operatorname{e}}^{x}}+{{\operatorname{e}}^{-x}})\]is increasing for all \[x>0.\]                 Statement 2: The functions \[{{x}^{2}}{{e}^{x}}\] and \[{{x}^{2}}{{e}^{-x}}\] are increasing for all \[x>0\] and the sum of two increasing functions in any interval (a, b) is an increasing function in (a, b).     JEE Main  Online Paper (Held On 22 April 2013)

    A)  Statement 1 is false; Statement 2 is true.

    B)  Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

    C)  Statement 1 is true; Statement 2 is false.

    D)  Statement 1 is true; Statement 2 is true; Statement 2 is a correct explanation for Statement 1. 

    Correct Answer: C

    Solution :

     Let \[y={{x}^{2}}.{{e}^{-x}}\] For increasing function, \[\frac{dy}{dx}>0\Rightarrow x[(2-x){{e}^{-x}}]>0\] \[\because \]\[x>0,\]  \[\therefore \]\[(2-x){{e}^{-x}}>0\] \[\Rightarrow \] \[(2-x)\frac{1}{{{e}^{x}}}>0\] For \[0<x<2,(2-x)<0\] \[\therefore \]\[\frac{1}{{{e}^{x}}}<0\],but it is not possible Hence the statement-2 is false.


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