Exploring the Relationship Between Derivatives and Uniform Continuity

[JasonJo]

Homework Statement

Suppose f and g are differentiable on R, and f(a) = g(a) and f'(x) <= g'(x) for all x >= a. Show that f(x) <= g(x) for all x >= a. Give a physical interpretation of this result.

Also, using the Mean Value Theorem:
(a) Let f: R --> R be a differentiable function. Suppose that its derivative f'(x) is bounded Prove that f is uniformly continuous.
(b) Let f: R --> R be a differentiable function. Suppose that lim (x --> infinity) f'(x) = infinity. Show that f cannot be uniformly continuous.
(c) Let g(x) = (x)^1/2 show that g'(x) is unbounded on (0,1] but g(x) is uniformly continuous on [0,1].

Homework Equations

The Attempt at a Solution

I don't know how to quite formulate the inequalities, any help?

 

[AKG]

Consider the function f-g.

 

[JasonJo]

i still don't see it, any more hints? thanks

 

[AKG]

(f-g)(a) = 0, (f-g)'(x) < 0 for all x > a. Use MVT to prove that (f-g)(x) < 0 for all x > a.