May someone please show me how to proof the following:
Let f be differentiable on I. Show that if f is monotone increasing on I, then
f '(x) >= 0 for all x in I.
Note: >= is greater or equal to.
My idea:
Suppose that f is monotone increasing, that is f(x) <= f(c) for all x such that x<c. ... Then I know that after, we have to somehow say that [f(x)-f(c)]/(x-c) >= 0.
Can someone explain more and do we need any theorem to support the proof?
Thank you .
If at some point in , then at that point (this uses the definition of having a derivative at that point):
but we have just shown that as approaches from the left (using the assumption that is monotone increasing and the existance of the derivative at the point):
which is a contradiction.
The only properties it uses are that is monotone increasing and the definition of a derivative.
CB