# Thread: No. of real roots.

1. ## No. of real roots.

Let $\displaystyle f(x)$ be a polynomial of degree $\displaystyle n$, an odd positive integer, and has monotonic behaviour , then the number of real roots of the equation
$\displaystyle f(x)+f(2x)+......+f(nx) = \frac{1}{2} n(n+1)$ is equal to

2. I think the point is that $\displaystyle f(x)$ is monotonic means that $\displaystyle f(kx)$ 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?

3. Originally Posted by snowtea
I think the point is that $\displaystyle f(x)$ is monotonic means that $\displaystyle f(kx)$ 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.