# No. of real roots.

• January 8th 2011, 02:32 AM
jacks
No. of real roots.
Let $f(x)$ be a polynomial of degree $n$, an odd positive integer, and has monotonic behaviour , then the number of real roots of the equation
$f(x)+f(2x)+......+f(nx) = \frac{1}{2} n(n+1)$ is equal to
• January 8th 2011, 07:11 AM
snowtea
I think the point is that $f(x)$ is monotonic means that $f(kx)$ is monotonic in the same direction (increasing/decreasing).

The sum of of monotonic functions in the same direction is monotonic.

How many real roots can a monotonic function have?
E.g. how many times can a strictly increasing function intersect the x-axis?
• January 8th 2011, 07:12 PM
jacks
Quote:

Originally Posted by snowtea
I think the point is that $f(x)$ is monotonic means that $f(kx)$ is monotonic in the same direction (increasing/decreasing).

The sum of of monotonic functions in the same direction is monotonic.

How many real roots can a monotonic function have?
E.g. how many times can a strictly increasing function intersect the x-axis?

one solution.

Thanks.