Serlo: EN: Monotonic functions

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Monotony criterion

The monotony criterion is quite intuitive: if the derivative of a function (i.e. the slope) is positive, it goes up, if the derivative is negative, it goes down. Mathematically, if the derivative $f^{\prime }$ of a differentiable function $f$ is non-negative or (non-positive) on an interval $(a,b)$ , then $f$ is monotonously increasing (or decreasing) on $(a,b)$. If $f^{\prime }$ is even strictly positive (or negative) $(a,b)$, then $f$ is strictly monotonously increasing (or decreasing).

In the first case, even inversion of the statement is true: If a differentiable function is monotonously increasing on $[a,b]$, then $f^{\prime }(x)\ge 0$ and if the function is monotonously decreasing on $[a,b]$, then then $f^{\prime }(x)\ge 0$. However, the inversion does not hold true in the strict case, monotone functions do not always have $f^{\prime }(x)>0$ or $f^{\prime }(x)<0$ . For instance, $f(x)=x^3$ is strictly monotonous, but $f^{\prime }(0)=3\cdot 0^2=0$.

Proof

The four directions "$⟹$" follow from the mean value theorem. The two directions "$⟸$" follow by differentiability of the function:

Exercise

Necessary and sufficient criterion for strict monotony

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