Thank you for reporting, we will resolve it shortly
Q.
The minimum distance between a point on the curve $y=e^{x}$ and a point on the curve $y=\log _{e} x$ is
Application of Derivatives
Solution:
Since $y=e^{x}$ and $y=\log _{e} x$ are inverse to each other their graphs are symmetrical about $y=x$.
Minimum distance between the curves is the distance between the points on the curves where tangent is parallel to $y=x$ or slope of tangent is $1$ .
Now for $y= e ^{x}, y'=e^{x}$, for $e^{x}=1, x=0$, so the point on the curve $y=e^{x}$ is $(0,1)$ and symmetric point on the curve $y=\log _{e x}$ is $(1,0)$
Distance between these points is $\sqrt{2}$.
