monotonic function

The first derivative of a function, \( f'(x) \), reveals where the original function \( f(x) \) is increasing, decreasing, or remains monotonic over specific intervals.

The Beta distribution is a continuous probability distribution defined over the open interval \( (0, 1) \). In formal terms, the beta distribution is defined by the following probability density function:

\[
B(x; \alpha, \beta) =
\frac{x^{\alpha – 1}(1 – x)^{\beta – 1}}{B(\alpha, \beta)}
\quad 0 < x < 1 \]