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## 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?