Question

The co-ordinates of the point P on the graph of the function $y=e^{-|x|}$ where the portion of the tangent intercepted between the co-ordinate axes has the greatest area, is

• A. $(1,\frac{1}{e})$
• B. $(-1,\frac{1}{e})$
• C. $(e,e^{-e})$
• D. none

Answer 1

Answer: (A), (B)

$(1,\frac{1}{e})$
$(-1,\frac{1}{e})$

$y=e^{-|x|}$
$y=e^{-x},x>0$, $e^x,x<0$
For $y=e^{-x}$ let point $(x_1,y_1)$ lies on it
So, slope is $\frac{dy}{dx}=-e^{-x_1}$
Therefore equation of tangent is $\frac{dy}{dx}=-e^{-x_1}$
Hence X intercept is $(1+x_1)$ and Y intercepts is $e^{-x_1}(1+x_1)$
Area so formed is $A=\frac{1}{2}{e}^{-x_1}{(1+x_1)}^2$
And this is maximum at $x_1=1\Rightarrow y_1=\frac{1}{e}$
Similarly for $y=e^x$ we get $x_1=-1\Rightarrow y_1=\frac{1}{e}$
Hence, option 'B' correct