Answer
$x=\dfrac{1}{\sqrt 2}$ and $y=\dfrac{1}{\sqrt e}$ and $A=\frac{1}{\sqrt{2e}}$
Work Step by Step
Let the area of a rectangle be $a=xy$
Here, $a= xe^{-x^2}$
Then $a'=e^{-x^2}-2x^2e^{-x^2}=0$
This implies that $x=\dfrac{1}{\sqrt 2}$
Now, $y=e^{-x^2}=e^{(-1/\sqrt 2)^2}=\dfrac{1}{\sqrt e}$
Hence, $x=\dfrac{1}{\sqrt 2}$ and $y=\dfrac{1}{\sqrt e}$
The maximum area is: $A=x\times y=\frac{1}{\sqrt{2e}}$
