Suppose thatf : [0, 1]--->R is continuously differentiable.
(a) Show that there is some number M such that |f'(x)|< M for all x.
I understand mean value therorem but use it to solve this problem. Can you give me some hints please?
Thanks
Suppose thatf : [0, 1]--->R is continuously differentiable.
(a) Show that there is some number M such that |f'(x)|< M for all x.
I understand mean value therorem but use it to solve this problem. Can you give me some hints please?
Thanks
A continuously differentiable function on a compact set means that it's continuous (in this case) on $\displaystyle [0,1]$ and with a continuous derivative on $\displaystyle [0,1].$ Hence, since the derivative is continuous on a compact set, it's bounded.
From lecture notes: By the boundedness principle, a continuous function on a closed interval attains its maximum and minimum.
Therefore there exists a number M such that f(x)<=M for all x belong to [0,1]. How do i apply this to $\displaystyle |f'|$
I think i have to do something like.... how do i complete the answer? thanks
$\displaystyle 0<|f'|<max(m1,m2)$