monotonicity criterion
Suppose that $f:[a,b]\to \mathbb{R}$ is a function^{} which is continuous^{} on $[a,b]$ and differentiable^{} on $(a,b)$.
Then the following relations hold.

1.
${f}^{\prime}(x)\ge 0$ for all $x\in (a,b)$ $\iff $ $f$ is an increasing function on $[a,b]$;

2.
${f}^{\prime}(x)\le 0$ for all $x\in (a,b)$ $\iff $ $f$ is a decreasing function on $[a,b]$;

3.
${f}^{\prime}(x)>0$ for all $x\in (a,b)$ $\Rightarrow $ $f$ is a strictly increasing function on $[a,b]$;

4.
$$ for all $x\in (a,b)$ $\Rightarrow $ $f$ is a strictly decreasing function on $[a,b]$.
Notice that the third and fourth statement cannot be inverted. As an example consider the function $f:[1,1]\to \mathbb{R}$, $f(x)={x}^{3}$. This is a strictly increasing function, but ${f}^{\prime}(0)=0$.
Title  monotonicity criterion 

Canonical name  MonotonicityCriterion 
Date of creation  20130322 13:45:12 
Last modified on  20130322 13:45:12 
Owner  paolini (1187) 
Last modified by  paolini (1187) 
Numerical id  7 
Author  paolini (1187) 
Entry type  Theorem 
Classification  msc 26A06 