Question

Construct the graph of the given function and state its nature:

$y=2.e^{x^2-4x}$

• A. Increasing for $x>2$ and decreasing for $x<2$
• B. Increasing for $x<2$ and decreasing for $x>2$
• C. Increasing for $x>-2$ and decreasing for $x<-2$
• D. Increasing for $x<-2$ and decreasing for $x>-2$

Answer 1

The correct option is A Increasing for $x>2$ and decreasing for $x<2$

Given function is $y=2.e^{x^2-4x}$
Hence, $y=f\left(x\right)$ foes not cut the $x$-axis at any point.
Now $f\left(x\right)=2.e^{x^2-4x}$
$=2.e^{(x-2{)}^2-4}$
$=2.e^{-4}.e^{(x-2{)}^2}$
Hence, $f^{\prime }\left(x\right)>0$ implies
$2(x-2{)}^2(x-2)>0$
$(x-2{)}^3>0$
Hence, Increasing for $x>2$ and decreasing for $x<2$.