Derivative Conditions for a Monotonic (Increasing or Decreasing) Function

The link between a function's derivative and its monotonicity follows from the geometric meaning of the derivative established earlier in this chapter: at any point, $f'(x)$ is the slope of the tangent there. On the graph of an increasing function, every tangent drawn within the domain makes an acute angle of inclination with the $x$-axis (occasionally touching $0°$ or $90°$ at isolated points) — the curve is, informally, "upward sloping." On the graph of a decreasing function, every tangent instead makes an obtuse angle — the curve is "downward sloping."

This geometric picture gives two derivative tests:

Both conditions are sufficient but not necessary — a function can be increasing over its whole domain even though its derivative is zero (or undefined) at a few isolated points, so failing the strict inequality at one point does not, by itself, rule out monotonicity there.

Points where the derivative is zero or does not exist deserve special attention, because they mark where a curve can change character. An interior point $c$ of the domain of $f$ is called a critical point if $f$ is continuous at $c$ and either $f'(c) = 0$ or $f'(c)$ is not defined — this also covers any point where $f$ itself is discontinuous. At a critical point, the graph can do several different things: it can take a smooth turn where the tangent is horizontal (as at the vertex of $f(x) = (x-1)^2 + 2$, where $f'(1) = 0$); it can have a sharp corner, where no single tangent line exists; it can have a vertical tangent that also marks a point of inflexion, where the concavity of the curve changes; or it can have a cusp, a sharply pointed turn, as seen in $f(x) = x^{2/3}$ at the origin. …