Welcome to our community

Be a part of something great, join today!

Find the minimal distance

  • Thread starter
  • Admin
  • #1

anemone

MHB POTW Director
Staff member
Feb 14, 2012
3,678
What is the minimal distance between \(\displaystyle y=e^x\) and \(\displaystyle y=\ln x\)?
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,702
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 $\sqrt2$.