Question 50
Let $f: [0, \infty) \to (0, \infty)$ be a continuous function. Prove that:
a) if $n$ is a large enough integer, say, $n \gt n_0$, then $n \int_0^{x_n} f(t) \, dt = 1$ for a unique positive real number $x_n$;
b) the sequence $(nx_n)_{n \gt n_0}$ is convergent and evaluate its limit.
a) if $n$ is a large enough integer, say, $n \gt n_0$, then $n \int_0^{x_n} f(t) \, dt = 1$ for a unique positive real number $x_n$;
b) the sequence $(nx_n)_{n \gt n_0}$ is convergent and evaluate its limit.