monotonicity criterion

Suppose that $f\colon[a,b]\to\mathbb{R}$ is a function which is continuous on $[a,b]$ and differentiable on $(a,b)$ .

Then the following relations hold.

1.

$f^{\prime}(x)\geq 0$ for all $x\in(a,b)$ $\Leftrightarrow$ $f$ is an increasing function on $[a,b]$ ;
2.

$f^{\prime}(x)\leq 0$ for all $x\in(a,b)$ $\Leftrightarrow$ $f$ 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.

$f^{\prime}(x)<0$ 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 $f\colon[-1,1]\to\mathbb{R}$ , $f(x)=x^{3}$ . This is a strictly increasing function, but $f^{\prime}(0)=0$ .

Title	monotonicity criterion
Canonical name	MonotonicityCriterion
Date of creation	2013-03-22 13:45:12
Last modified on	2013-03-22 13:45:12
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	7
Author	paolini (1187)
Entry type	Theorem
Classification	msc 26A06