1. ## derivative/real analysis

Problem 1:

Let f(x) be a real valued function defined for all x >= 1, that satisfies
f(1) = 1 and
f'(x) = 1/[x^2 + (f(x))^2]
Prove that lim f(x) as x tends to infinity exists and is less than 1+ pi/4.

Problem 2:

Suppose that a continuously differentiable function f : IR --IR satisfies
f'(x) = g(f(x)) + h(x) for x in IR,
where the functions g, h : IR ---IR are C 00 (i.e. infinitely differentiable). Prove that
the function f is infinitely differentiable as well.

2. ## Re: derivative/real analysis

Problem 1
$f$ is increasing, and $f'(x)\leq \frac 1{1+x^2}$. Write $f(x)-f(1)=\int_1^xf'(t)dt$ to get the result.

Problem 2
You have to use (or show) the fact that the composition of two smooth functions is smooth, and the sum of two smooth functions is smooth.