# Prove existence of a real root.

Problem: If

$$C_0+\frac{C_1}{2}+\cdots + \frac{C_{n-1}}{n}+\frac{C_n}{n+1} =0,$$

where $C_0,...,C_n$ are real constants, prove that the equation

$$C_0+C_1x+\cdots +C_{n-1}x^{n-1}+C_nx^n=0$$

has at least one real root between $0$ and $1$.

Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 4.

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I really don't think that posting questions one-by-one from a given textbook chapter -- and then answering them within one minute of posting -- is considered good form. –  Jesse Madnick Jun 30 '12 at 6:39
That is, if you know the answer to the question you're asking, and what you really want is feedback, then I think your solution should be in the body of the question. –  Jesse Madnick Jun 30 '12 at 6:43

$$g(x)=C_0x+\frac{C_1}{2}x^2+\cdots + \frac{C_{n-1}}{n}x^n+\frac{C_n}{n+1}x^{n+1}$$
is an antiderivative for $f$. Note further that $g(0)=0$ and $g(1)=0$ by hypothesis. Then there exists $t\in(0,1)$ with $g'(t)=0$, that is, $f(t)=0$.