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.