Showing two sets have touching boundaries (function sets)

Question

Let $f: [0,1] \rightarrow \mathbb{R}$ be continuous on $[0,1]$ and differentiable on $(0,1)$. Suppose that $f(0)< 0 < f(1)$ and $f'(x) \neq 0$ for every $x \in (0,1)$. Let $S_{1} = \{ x \in [0,1]: f(x) > 0\}$ and $S_{2} = \{x \in [0,1]: f(x) < 0\}$. Prove that $\inf(S_{1}) = \sup(S_{2}).$

Answer 1

Hint: Rolle's theorem, together with the $f'(x)\neq 0$ hypothesis, tells you that $f(x)$ has at most one zero in $[0,1]$.