monotonicity criterion

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