Statement - 1: The function x^2 (e^x + e^ -x ) is increasing for … - Vidyadip

MCQ

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 $- 1.$
✓
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

Correct option: C.

Statement $-1$ is true; Statement $-2$ is false.

c
Let $y=x^{2} \cdot e^{-x}$

For increasing function,

$\frac{d y}{d x}>0 \Rightarrow x\left[(2-x) e^{-x}\right]>0$

$\because x>0, \therefore(2-x) e^{-x}>0$

$\Rightarrow(2-x) \frac{1}{e^{x}}>0$

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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