I'm stuck on this problem :

Suppose that we use the bisection method to find some roots of a continuous function $\displaystyle f(x)$ in the interval $\displaystyle [a,b]$ and that $\displaystyle f(a)f(b)<0$.

Suppose also that $\displaystyle x_i$, $\displaystyle i$ goes from $\displaystyle 1$ to $\displaystyle k$ are the roots of $\displaystyle f(x)$ in $\displaystyle [a,b]$, they are simple roots (what I understand by this is that they are not double roots) and that $\displaystyle x_1<x_2<...<x_k$.

1)Study the parity of $\displaystyle k$.

I will try to do the other questions of this problem once I understand how to do the 1).

Ohh... I think I just understood. By doing a draw, I notice that if the number of roots is odd then $\displaystyle f(a)f(b)<0$ and if it's even, $\displaystyle f(a)f(b)>0$. How can I prove that?