1. Let's suppose it has more than one (it clearly has at least one by Bolzano), which would be, at least 2.

$\displaystyle x_1$ and $\displaystyle x_2$ denote 2 of our roots

We are supposing that

$\displaystyle x_1\in(a,b)$ and $\displaystyle x_2\in(a,b)$ with $\displaystyle x_1\neq{x_2}$ and $\displaystyle f(x_1)=f(x_2)=0$

Then, by Rolle's Theorem, there exists $\displaystyle c\in{(x_1,x_2)}$ such that: $\displaystyle f'(c)=0$

But $\displaystyle f'(x)$ can't be 0 in (a,b)

Thus it is absurd to suppose it has more than 1 root