Question

Find the intervals of monotonocity for the following functions & reprsent your solution set the number line.

(i)$f\left(x\right)=2.e^{x^2-4x}$

(ii)$f\left(x\right)=\frac{e^x}{x}$

(iii)$f\left(x\right)=x^2{e}^{-x}$

Answer 1

Solution:

$1$

$f\left(x\right)=2.e^{x^2-4x}$

$f^{‘}\left(x\right)=2.e^{x^2-4x}(2x-4)$

Now $f^{‘}\left(x\right)=0\Rightarrow (2x-4)=0$

$\Rightarrow (x-2)=0\Rightarrow x=2$

Now $f\left(x\right)=(+)ve$ for all $x$ Increasing in all $R$

$2$

$f\left(x\right)=\frac{e^x}{x}$

$f^{‘}\left(x\right)=\frac{x{e}^x-e^x}{x^2}$

$=\frac{e^x(x-1)}{x^2}$

Now $f^{‘}\left(x\right)=0\Rightarrow x=1$

$f\left(x\right)$ increase on $[1,\infty )$

$f\left(x\right)$ decreases on $(-\infty ,0)U(0,1)$

$3$

$f\left(x\right)=x^2{e}^{-x}$

$f\left(x\right)=2x\left(e^{-x}\right)-e^{-x}\left(x^2\right)$

$=x{e}^{-x}(2-x)=0$

$\Rightarrow x=0$ or $x=2$

$f\left(x\right)$ increase on $(0,2)$

$f\left(x\right)$ decrease on $(\infty ,0)U(2,\infty )$