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-

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

s u p p o s e t h a t f i s c o n t i n u o u s a n d d i f f e r e n t i a b l e i n (a, b) i f f^{'} (x) = 0, \forall x \in (a, b) t h e n s h o w t h a t f i s c o n s t a n t o n [a, b] .

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

assum f is not consrante

\exists x, y suche that 1 ⩾ y \neq x ⩾ 0 f (x) \neq f (y)

use mean value theorem for f in [x, y] \Rightarrow

\exists c \in [x, y] such that

\frac{f (y) - f (x)}{y - x} = f^{'} (c) \Rightarrow f (y) - f (x) = (y - x) f^{'} (c)

but f^{'} (c) = 0 cause c \in [0, 1]

\Rightarrow f (y) - f (x) = 0 \Rightarrow f (y) = f (x) absurd \Rightarrow \forall (x, y) \in {[0, 1]}^{2} f (x) = f (y)

f constant

Commented by psyche last updated on 24/Oct/19

t h a n k s

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