Let f be differentiable and f strictly increasing on R and f

Let f be differentiable and f\' strictly increasing on R, and f(a) = f(b) for two points a

Solution

Let, some x so that

f(x)>=f(a)=f(b)

By Mean Value Theorem there is some c1 in (a,x) so that

f\'(c1)=(f(x)-f(a))/(b-a)>=0

Also there is some c2 in (x,b) so that

f\'(c2)=(f(b)-f(x))/(b-a)<=0

But, c2>c1 and f\' is stricly increasing so we have a contradiction

Hence, f(x)<f(a)=f(b) for all x in (a,b)