Question Number 71816 by psyche last updated on 20/Oct/19

suppose that f is continuous and differentiable in (a,b) if f′(x) =0 ,∀ x∈(a,b) then show that f is constant on [a,b].

Answered by mind is power last updated on 20/Oct/19

assum f is not consrante  ∃x,y suche that 1≥y≠x≥0   f(x)≠f(y)  use  mean value theorem for f in [x,y]⇒  ∃c∈[x,y] such that  ((f(y)−f(x))/(y−x))=f′(c)⇒f(y)−f(x)=(y−x)f′(c)  but f′(c)=0 cause c∈[0,1]  ⇒f(y)−f(x)=0⇒f(y)=f(x) absurd⇒∀(x,y)∈[0,1]^2  f(x)=f(y)  f constant

Commented bypsyche last updated on 24/Oct/19