Analysis Homework Help

In summary, the conversation discusses the relationship between differentiable functions f and g on the real numbers, and their derivatives. It is shown that if f'(x) is always less than or equal to g'(x) for all x greater than or equal to a, then f(x) will also be less than or equal to g(x) for all x greater than or equal to a. This result has a physical interpretation. Additionally, using the Mean Value Theorem, it is proven that if a differentiable function's derivative is bounded, then the function itself is uniformly continuous. On the other hand, if the limit as x approaches infinity of the derivative is infinity, then the function cannot be uniformly continuous. Lastly, an example is given
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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?
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  • #2
Consider the function f-g.
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i still don't see it, any more hints? thanks
  • #4
(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.

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