Question

The coordinates of the point P(x, y) of $y=e^{-|x|}$ so that the area formed by the coordinate axes and the tangent at P is greatest, are

• A. (1, 1/e)
• B. (1, -1/e)
• C. (1, e)
• D. (-1, e)

Answer 1

Answer: (A)

Grayph of $y=e^{-|x|}$ is symmetrical about y-axis, so we coxnsider $x\ge 0$, then $y=e^{-x}\Rightarrow \frac{dy}{dx}=-e^x$

Tangent at P is $Y-y=-e^{-x}\left(X-x\right)$
Its x-intercept $=x+y{e}^x$ and y-intercept $=y+x{e}^{-x}$
So area $A=\frac{1}{2}\left(x+y{e}^x\right)\left(y+x{e}^{-x}\right)$
$=\frac{1}{2}\left(x+1\right)\left(1+x\right)e^{-x}\left[\because y=e^{-x}\right]$
$\frac{1}{2}{\left(1+x\right)}^2{e}^{-x}$
$\frac{dA}{dx}=\left(1+x\right)e^{-x}-\frac{1}{2}{\left(1+x\right)}^2{e}^{-x}=\frac{1}{2}\left(x+1\right)\left(1-x\right)e^{-x}$
where A is maximum if $x=1$
So P is $\left(1,e^{-1}\right)$ Due to symmetry, there is another point $\left(-1,e^{-1}\right)$