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.