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
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
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?