Derivatives and continuity of one variable functions 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 $f'(c) = 0$.
 A: 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.
