# Functions 2

## Homework Statement

Let f(x) be a pllynomial 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(3x)...+ f(nx)= n(n+1)/2

## The Attempt at a Solution

This seems like the summation of 1+2+3+...+n, and that would be a root of the equation, but I dont know if thats the only one.

AKG
Homework Helper
Let f(x) be a pllynomial 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(3x)...+ f(nx)= n(n+1)/2
You didn't finish asking a question. Do you mean that you have to show that the number of solutions of

f(x) + ... + f(nx) = 0

is n(n+1)/2, or are you asked to find the number of solutions of the equation

f(x)+ f(2x) + f(3x)...+ f(nx)= n(n+1)/2?

HallsofIvy
Homework Helper
Not to mention: what do you mean by "has monotonic behavior"? Do you mean that it is monotonic for all x? If so then f(x)= 0 has exactly one root.

You have to find the number of roots of the equation. And it is monotonic for all x. How did you get f(x)=0 has only one root?

AKG
Homework Helper
So you have to find the number of x-values such that

f(x) + ... + f(nx) = n(n+1)/2

Right? f(x)=0 has only one root because f is monotonic, so once it crosses the x-axis once, it will never cross again. Of course, it must cross the x-axis at least once because f is an odd-degree polynomial.

HallsofIvy
Homework Helper
I would recommend looking at simple cases first. It is easy to show that if f(x)= ax+ b, there is exactly one root to that equation, but what if f(x)= x3 or variations on that?

Since f(x) is monotonic for all x, then it has only one solution. I.e, it only cuts the line y=0 at one point. Let f(x) be a function of the type $$f(x)=a_nx^n+a_{n-1}x^{n-1}...$$
Then f(x)+f(2x)+f(3x)+...+f(nx) will also be a function of the type $$F(x)=A_nx^n+A_{n-1}x^{n-1}...$$ which will also be monotonic, and hence have only one solution.

Is that right?

AKG
Homework Helper
$x \mapsto f(x)$ is either monotonic increasing or monotonic decreasing. Without loss of generality, suppose it is monotonic increasing. Then what can you say about the functions $x \mapsto f(kx)$ where k is some positive integer?

f(kx) is also monotonically increasing or decreasing. Graphically, if k<1, then the graph expands, and if k>1, then the graph contracts, but the graph remains basically the same.

AKG