Increasing and Decreasing Functions (Monotonic Functions)

A function's derivative doesn't just give the slope at a point — it also tells you whether the function is climbing or falling as $x$ increases, which is the idea of monotonicity.

A real function $y = f(x)$ is called increasing on an open interval $(a, b)$ if the value of $f$ increases as $x$ increases — graphically, the curve rises as you move from left to right. The interval $(a,b)$ here can be a bounded interval between two real numbers, or an unbounded one such as $(a, \infty)$, $(-\infty, b)$, or all of $\mathbb{R} = (-\infty,\infty)$. Analytically, $f$ is increasing on $(a,b)$ if, for every pair of points $x_1, x_2 \in (a,b)$,

A function is decreasing on $(a,b)$ if instead the value of $f$ falls as $x$ increases — the curve falls left to right — which analytically means