Question 22
Let $f:[a,b]\to\mathbb R$ be a function with the Darboux property such that $f(a)f(b)\lt 0$. Prove that there exist $\alpha,\beta$ with $a\lt \alpha\lt \beta\lt b$ and
$$
f(\alpha)+f(\beta)=f(\alpha)f(\beta).
$$
$$
f(\alpha)+f(\beta)=f(\alpha)f(\beta).
$$