Monotonicity of Functions

Monotonicity describes a function's behaviour of increasing or decreasing.

Definition 7.4 (Increasing function). $f(x)$ is increasing on an interval $I$ if $a<b\Rightarrow f(a)\le f(b)$ for all $a,b\in I$.

Definition 7.5 (Decreasing function). $f(x)$ is decreasing on $I$ if $a<b\Rightarrow f(a)\ge f(b)$ for all $a,b\in I$.

$f(x)=x$ is increasing on the entire real line; $f(x)=-x$ is decreasing on the entire real line. A function may be increasing on one interval and decreasing on another — e.g. $f(x)=|x|$ is decreasing on $(-\infty,0]$ and increasing on $[0,\infty)$. For a general function, determining monotonicity by inspection quickly becomes impossible, which is exactly what the following theorem resolves.

Theorem 7.7. If $f(x)$ is differentiable on the open interval $(a,b)$:

1. If $\dfrac{d}{dx}\big(f(x)\big)\ge0$ for all $x\in(a,b)$, then $f(x)$ is increasing on $(a,b)$.
2. If $\dfrac{d}{dx}\big(f(x)\big)>0$ for all $x\in(a,b)$, then $f(x)$ is strictly increasing on $(a,b)$. (The proof follows from Theorem 7.3, LMVT.)
3. $f(x)$ is decreasing on $(a,b)$ if $\dfrac{d}{dx}\big(f(x)\big)\le0$ for all $x\in(a,b)$.
4. $f(x)$ is strictly decreasing on $(a,b)$ if $\dfrac{d}{dx}\big(f(x)\big)<0$ for all $x\in(a,b)$.

Definition 7.6 (Stationary point). A stationary point $\big(x_0,f(x_0)\big)$ of a differentiable $f(x)$ is where $f'(x_0)=0$. …