The Mean Value Theorem and Fixed Points

all right thank you so much for watching so someone couple of months ago asked me to do a nice application of the mean value theorem and I thought well this is my my favorite application of it so let me present you two that has to do with fixed points which really means if you

want crossing the line y equals 2x so suppose your function you know from 0 1 to 0 1 in other words a function with domain 0 1 and that's never bigger than 1 it's always between 0 and 1 and the question is can you draw this without crossing the line y equals 2x oh of

course because it can't cross the line here it has to start somewhere here and then while it's tried to do this without doing that and then of course you don't want everyone to be 1 because it crosses the line and the question is how do you do that oh no it's not impossible and in

fact let me show that it's not possible and let me show a little bit more so theorem some kind of fixed point theorem suppose F is from 0 1 to 0 1 and if derivative is never equal to 1 so an F prime of X is never equal to 1 for all X

in 0 1 then F must cross this line exactly once so it has to cross it at least once and it cannot cross it twice or more F must have exactly one fixed point in other words a fixed point just means that F crosses this line which means that f of X equals 2x must have exactly

one solution I always get very excited about this problem but whenever I ask it to my students they're never very excited about this but I hope you won't disappoint me because in fact it's a very nice application of both of the value theorems one the extreme value theorem and one sorry not sorry