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Any help with these problems. Thanks in advance.

Problem 1:

Let $f(x)$ be a real valued function defined for all $x \geq 1$, that satisfies $f(1) = 1$ and $\displaystyle f'(x) = \frac{1}{x^2 + (f(x))^2}$ Prove that $\lim_{x \to \infty} f(x)$ exists and is less than $1+ \pi/4$.

Problem 2:

Suppose that a continuously differentiable function $f : \Bbb{R} \to \Bbb{R}$ satisfies $f'(x) = g(f(x)) + h(x)$ for $x \in \Bbb{R}$, where the functions $g, h : \Bbb{R} \to \Bbb{R}$ are $C^\infty$ (i.e. infinitely differentiable). Prove that the function $f$ is infinitely differentiable as well.

Problem 3:

Prove that if $f : [0,1) \to \Bbb{R}$ is nonnegative, integrable, and uniformly continuous, then $\lim_{x \to \infty} f(x) =0$.

Problem 4:

Suppose that a differentiable function $f : \Bbb{R} \to \Bbb{R}$ and its derivative $f'$ have no common zeros. Prove that $f$ has only finitely many zeros in $[0, 1]$.

Problem 5:

Suppose that $f : [0,\infty)\to \Bbb{R}$ is continuous on $[0,\infty)$, differentiable on $(0,\infty), f(0) = 0$, and $\lim_{x \to \infty} f(x) = 0$. Prove that there exists a point $c$ in $(0,\infty)$ such that $f'(c) = 0$.

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I guess that the intervals $[0,1)$ in the last problem should be replaced with $[0,\infty)$. I added $\LaTeX$ formatting, but next time you should try and post your question using the formatting required on the site. Please read the FAQ. Don't post again questions on which you did not try and do yourself. You just gave a list of problems, probably homework, with no mention of your work. –  Beni Bogosel Oct 16 '11 at 19:53
    
Yes, you're right. Sorry for the inconvenience this typo might cause. –  M.Krov Oct 16 '11 at 20:29
    
By the way, just to clarify the status of the posted topic: this is not a homework. It seems that since I posted 5 problems at a time that it is a homework, but it's not. It's just a set of practice problems that I am using for my preparation. –  M.Krov Oct 16 '11 at 20:33
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Yeah, but if you are preparing for something, you are trying to solve the problems by yourself before posting them here. Writing what you've done helps us give you the right hint to go on and continue your proof. Maybe you're not far from the solution. If you are the one who reaches the end of it, and you don't just read some solutions given by others, then you will learn more from solving these problems. What you find by yourself is more valuable for practicing your skill than reading a book. –  Beni Bogosel Oct 16 '11 at 20:49
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1 Answer

up vote 2 down vote accepted

For problem 1. You have two facts: $f'(x)>0$ which means that $f$ is increasing and therefore it has a limit as $x \to \infty$. Secondly, notice that if $f$ is increasing then $$ f'(x) \leq \frac{1}{1+x^2}$$. Integrate from $0$ to $\infty$ and you will get the desired result.

The second problem seems easy by induction. First note that $f'$ is differentiable, since it is the result of compositions and operations with differentiable functions. Calculate $f''$ from that formula, and you will get a formula with $g,g',h,h',f,f'$ which are all differentiable. By induction it follows that $f$ is $C^\infty$.

The third problem is Barbalat's Lemma. You can find a proof on my blog: http://mathproblems123.wordpress.com/2009/10/01/barbalats-lemma/

For the fourth problem, argue by contradiction. If there are infinitely many zeros for $f$ then they have an accumulation point $c$. Try and prove that $f'(c)=0$.

For the fifth problem, try and find a function which is $C^\infty$, increasing, and maps $[0,1]$ to $[0,\infty]$. Build the function $g=f \circ \phi$ and apply intermediate value theorem.

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For problem 2: Yes, I can easily find a recursive formula of the nth derivative of f in terms of f', f, nth derivatives of g and h, and use the fact that g,h are infinitely differentiable and their derivatives are continuous on R. Thanks for this useful hint. –  M.Krov Oct 16 '11 at 22:13
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