Find the minimal distance

[anemone]

What is the minimal distance between \(\displaystyle y=e^x\) and \(\displaystyle y=\ln x\)?

 

[Opalg]

What is the minimal distance between \(\displaystyle y=e^x\) and \(\displaystyle y=\ln x\)?

The graphs of those two functions are mirror images of each other in the line $y=x$. So the minimal distance between them will be the distance between the points where they come closest to that line. That will occur at the points where the tangents to the graphs are parallel to the line (which has slope $1$, of course), in other words at the points $(0,1)$ (on the exponential) and $(1,0)$ (on the logarithm). The minimal distance is therefore the distance between those two points, which is $\sqrt2$.