Thread: derivatives of functions

I appreciate if you can help me guys with any of these problems. They are good for practice and review of derivatives.

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.

problem 3:

Prove that if f : [0,1)--- IR is nonnegative, integrable, and uniformly
continuous, then lim f(x) =0 as x tends to 00.

Problem 4:

Suppose that a differentiable function f : IR ---IR and its derivative f'
have no common zeros. Prove that f has only finitely many zeros in [0, 1].

Problem 5:

Suppose that f : [0,1)---IR is continuous on [0,1), differentiable on
(0,00), f(0) = 0, and lim f(x) = 0 as x tends to 00. Prove that there exists a point c in (0,00) such
that f'(c) = 0.

Originally Posted by skybluesea2010

I appreciate if you can help me guys with any of these problems. They are good for practice and review of derivatives.

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.

problem 3:

Prove that if f : [0,1)--- IR is nonnegative, integrable, and uniformly
continuous, then lim f(x) =0 as x tends to 00.

Problem 4:

Suppose that a differentiable function f : IR ---IR and its derivative f'
have no common zeros. Prove that f has only finitely many zeros in [0, 1].

Problem 5:

Suppose that f : [0,1)---IR is continuous on [0,1), differentiable on
(0,00), f(0) = 0, and lim f(x) = 0 as x tends to 00. Prove that there exists a point c in (0,00) such
that f'(c) = 0.

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