Question

The largest area of a rectangle which has one side on the $x$-axis and the two vertices on the curve $y=e^{-x^{2}}$ is

• A. $\sqrt{2} \,e ^{-1 / 2}$
• B. $2 \,e ^{-1 / 2}$
• C. $e^{-1 / 2}$
• D. none

Answer 1

Answer: (A)

Let $A$ be area $A=(2 x)\left(e^{-x^{2}}\right), x > 0$

$\frac{ d A }{ dx }=-2\left( x +\frac{1}{\sqrt{2}}\right)\left( x -\frac{1}{\sqrt{2}}\right) e ^{- x ^{2}}$
At $x=\frac{1}{\sqrt{2}}$, A is maximum.
Largest area is $2 \frac{1}{\sqrt{2}} e ^{-1 / 2}$