Problems of mathematic's

Problem 1

Find all the pairs of positive numbers (x, y) for which the function

has a minimum. Find the smallest possible minimum.

Problem 2

Prove that if the function f: R R satisfies the inequalities
f(x) x; f(x+y) f(x) + f(y)
for all real (x, y), then
f(x) x.

Problem 3

Prove that

Problem 4

Prove that the function g(x) = f(x) + f(ax) is periodic if and
only if a Q.
The function f(x) is given by

Problem 5

f(x) and g(x) are differentiable, non-constant functions. Prove
that if
a) f'(0)=0;
b) f(x+y) = f(x)f(y) - g(x)g(y);
c) g(x+y) = g(x)f(y) + g(y)f(x),
then
f²(x) + g²(x) = 1

Problem 6

Prove that if any function f: R R satisfies
one of these conditions, then it satisfies the other:
f(x+y) = f(x) + f(y);
f(xy+x+y) = f(xy) + f(x) + f(y).

Problem 7

Prove that if the non-constant function f: R R satisfies
the equation
f(x)f(y) = f(x+y)
for all real x and y and is differentiable for x = 0, then f(x) is
infinitely differentiable for all real x.

Problem 8

For any functions f, g: [0; 1] [0; 1], prove that if f
increases monotonically, then

Problem 9

Find

Problem 10

The differentiable functions f(x) and g(x) defined on [0; 1]
are such that
a) f(0) = f(1) = 1;
b) 153 f'(g(x)) + 41f(g'(x)) 0 for all x [0; 1].
Prove that g(1) g(0).