• # 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.

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.