# 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. 1.

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

2. 2.

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

3. 3.

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

4. 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 MonotonicityCriterion 2013-03-22 13:45:12 2013-03-22 13:45:12 paolini (1187) paolini (1187) 7 paolini (1187) Theorem msc 26A06