TITLE: Prove that the equation: $c_0+c_1x+\ldots+c_nx^n=0$ has a real solution between 0 and 1.
QUESTION [11 upvotes]: Let $c_0,c_1,c_2,\ldots ,c_n$ be constants such that :
$$c_0+\frac{c_1}{2}+\ldots+\frac{c_{n-1}}{n}+\frac{c_n}{n+1}=0$$
I have to prove that the equation:
$$c_0+c_1x+\ldots+c_nx^n=0$$
Has a real solution between 0 and 1.
Didn't know how to start...I thought that maybe I could use something about derivative...But I'm lost...
Any help,much appreciated!!!

REPLY [11 votes]: This is an exercise from the differentiation chapter in Baby Rudin, and knowledge of integration is not expected. You can solve it without integrals using the following function:
$$
f(x) = c_0x+\frac{c_1x^2}{2}+\ldots+\frac{c_{n-1}x^n}{n}+\frac{c_nx^{n+1}}{n+1}
$$
What value does it take at $0$ and $1$? What value must the derivative take in $(0, 1)$ then?