Question

Number of roots of the equation $x^2\cdot e^{2-|x|}=1$ is -

• A. 2
• B. 4
• C. 6
• D. infinite

Answer 1

Answer: (B)

$x^2{e}^{2-|x|}-1=0$
$f(x)=\frac{e^2{x}^2}{e^x}-1\forall x\ge 0$
$f^{\prime }(x)=e^2\frac{\left\{2x{e}^x-x^2{e}^x\right\}}{e^{2x}}=\frac{e^2\cdot x(2-x)}{e^x}$

$\Rightarrow f$ increases in $(0,2),f$ decreases in $(2,\infty )$
Also $f(0)<0\&f(2)>0\Rightarrow$ Exactly one root in $(0,2)$
$\underset{x\to \infty }{\mathrm{Lim}}f(x)<0$
$\Rightarrow$ exactly one root in $(2,\infty )$
$\Rightarrow$ exactly 2 roots in $(0,\infty )$
$\Rightarrow$ equation has exactly 4 roots
$\because f(x)$ is even function.