Please write STEP by step The last person who did it wrote s

Please write STEP by step. The last person who did it wrote sloppy and wouldn\'t correct it. IF YOU CANNOT RESPOND PLEASE LEAVE FOR SOMEONE ELSE. Please write all steps from beginning to end and if you\'re writing please write neat and legible. Don\'t just give the answer write each step. Thank you 20 1. Let f be a continuous differentiable function on R. Suppose that if\'()Sr

Solution

Let, x=f(x) have more than one distinct solutions

Let, u,v be two of them

So that f(u)=u,f(v)=v

Without loss of generality we assume: u<v

By Mean Value theorem tehre is some c in (u,v) so that

f\'(c)=(f(v)-f(u))/(v-u)=(v-u)/(v-u)=1

So,

f\'(c)=1

But, |f\'(x)|<1 for all x

So a contradiction

Hence, f(x) cannot have more than one fixed point ie f(x)=x solution must be unique