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.
Solution 1:
Note that
$$g(x)=C_0x+\frac{C_1}{2}x^2+\cdots + \frac{C_{n-1}}{n}x^n+\frac{C_n}{n+1}x^{n+1} $$
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, $f(t)=0$.