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Q.
Number of roots of the equation $x^2 \cdot e^{2-|x|}=1$ is -
Application of Derivatives
Solution:
$x ^2 e^{2-| x |}-1=0$
$f ( x )=\frac{e^2 x ^2}{e^{ x }}-1 \forall x \geq 0$
$f ^{\prime}( x )=e^2 \frac{\left\{2 x e^{ x }- x ^2 e^{ x }\right\}}{e^{2 x }}=\frac{e^2 \cdot x (2- x )}{e^{ x }}$
$\Rightarrow f$ increases in $(0,2), f$ decreases in $(2, \infty)$
Also $f (0)<0 \& f (2)>0 \Rightarrow$ Exactly one root in $(0,2)$
$\underset{x\rightarrow \infty}{\text{Lim}} f ( x )<0$
$\Rightarrow$ 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.
