they sign L as the minimum of f'(x)
and they sign l as the maximum of f'(x)
then they say that k=max{|f(L)|,|f(l)|}
but its not true
i am given that 0=<k<1
and i dont know if i get the same values
why
k=max{|f(L)|,|f(l)|} equals 0=<k<1
??
f is differentiable continuously(which means that f(x) and f'(x) are continues and differentiable) on [a,b]
suppose that |f'(x)|<1 for x in [a,b]
prove that there is 0<=k<1 so there is x1,x2 in [a,b]
|f(x1)-f(x2)|<=k|x1-x2|
??
i started from the data that i was given and i know that f(x) is differentiable
so
i use
|f'(x)|<1
so
so f(x) is continues and differentiable
so i use mvt on x1 and x2
and i combine that with
|f'(x)|<1
so
what now?