Question

Statement-1: The function $x^2(e^x+e^{-x})$ 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 $(a,b)$.

• 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

Answer 1

Answer: (C)

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