DElta Epsilon Proof Help please

[jim1027]

Hi, I need to do this proof and i am very lost, Can anyone show me how to do it.

A function f(x) is said to be increasing at a point x0 in its domain if there exists some ?>0 such thatf(x)?f(x0)forx0 ?x?x0+?,andf(x)?f(x0)for x0 ? ? ? x ? x0. Using the definition of the derivative, show that if f?(x0) > 0, then f is increasing at x0, that is, there exists some ? > 0 such that the above inequalities hold.

 

[DrMike]

The definition of the derivative is

\(\displaystyle f(x_0)=\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}\)

I'd suggest writing this into the eps-delta definition of the limit..., ie,

for all eps>0, there exists del>0 such that bla bla bla implies |something - f'(x_0)|<eps.

Choose eps equal, say, f'(x_0)/2. This is >0, since f'(x_0)>0. Why f'(x_0)/2 ? Because I want to deduce that the 'something' above is >0 also. This can probably be rearranged to give you what you want.