Question

Find the intervals of monotonicity for the following functions & represent your solution set on the number line. $f\left(x\right)=2.e^{x^2-4x}$

• A. $I$ in $(3,\infty )$ & D in $(-\infty ,3)$
• B. $I$ in $(2,\infty )$ & D in $(-\infty ,2)$
• C. $I$ in $(1,\infty )$ & D in $(-\infty ,1)$
• D. $I$ in $(4,\infty )$ & D in $(-\infty ,4)$

Answer 1

The correct option is B $I$ in $(2,\infty )$ & D in $(-\infty ,2)$

Given, $f\left(x\right)=2.e^{x^2-4x}$
$=2.e^{(x-2{)}^2-4}$
$=2.e^{-4}.e^{(x-2{)}^2}$
Hence, $f^{\prime }\left(x\right)>0$ implies $2(x-2{)}^2(x-2)>0$
$\Rightarrow (x-2{)}^3>0$
Hence, Increasing for $x>2$ and decreasing for $x<2$.