Question

Statement-1: The function $x^2 (e^x + e^{-x})$ is increasing for all $x > 0.$
Statement-2: The functions $x^2e^-{x}$ and $x^2e^{-x}$ 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.