Question 15
a) Let $a, b \in \mathbb{R}$ be two real numbers, with $a \lt b$, and $f : [a, b] \to \mathbb{R}$ a strict monotone function such that $\int_a^b f(x) dx = 0$. Show that $f(a) \cdot f(b) \lt 0$.
b) Determine the convergent sequences $(a_n)_{n \ge 1}$ of real numbers, for which there is a strict monotone function $f : \mathbb{R} \to \mathbb{R}$ such that
$$
\int_{a_{n-1}}^{a_n} f(x) \, dx = \int_{a_n}^{a_{n+1}} f(x) \, dx, \quad \text{for any } n \in \mathbb{N}, n \ge 2.
$$
b) Determine the convergent sequences $(a_n)_{n \ge 1}$ of real numbers, for which there is a strict monotone function $f : \mathbb{R} \to \mathbb{R}$ such that
$$
\int_{a_{n-1}}^{a_n} f(x) \, dx = \int_{a_n}^{a_{n+1}} f(x) \, dx, \quad \text{for any } n \in \mathbb{N}, n \ge 2.
$$