Derivative and monotonicity

One of the main applications of the Mean value theorem is getting information about the monotonicity of a function based on its derivative. Some information actually follows right from the definition of the derivative, namely the information about local phenomena (including local extrema). We start with it, later we move on to global phenomena.

Local monotonicity and extrema

Theorem (derivative and local monotonicity).
Let a be a point from the domain of a function f, assume that f has a derivative there.
If f is increasing or non-decreasing at a, then the derivative at a satisfies f ′(a) ≥ 0.
If f is decreasing or non-increasing at a, then the derivative at a satisfies f ′(a) ≤ 0.

If the derivative at a satisfies f ′(a) > 0, then f is increasing at a.
If the derivative at a satisfies f ′(a) < 0, then f is decreasing at a.

You probably noticed that the first two statements are unnecessarily complicated. Since an increasing function is also non-desreasing, it was enough to put "non-decreasing" in the first statement and "non-decreasing" in the second statement and they would automatically apply also to strict monotonicity. However, the way we wrote it is not incorrect, it just looks clumsy, and we had a good reason for it. We did it to suggest that these statements are the best we have. For instance, you might wonder: If "non-decreasing" gives non-negative derivative, would "increasing" give f ′(a) > 0? The answer is in the negative, this is not true in general. A simple counterexample to such a claim: The function f (x) = x³ is increasing at 0 (see e.g. Powers in Functions - Theory - Elementary functions), but f ′(0) = 0. Obviously, the paired "decreasing f hence negative f ′" statement is also not true in general.

In the second part of the theorem there is one possible place to change the statements. Would it be possible to claim that if f ′(a) ≥ 0, then f is non-decreasing at a? Again, this is not possible, for instance f (x) = −x³ satisfies f ′(0) ≥ 0 as f ′(0) = 0, but the function is decreasing at 0.

An analogous theorem ties together one-sided monotonicity and one-sided derivatives.

Another local feature is local extrema.

Theorem (derivative and local extrema).
Let a be a point from the domain of a function f, assume that f has a derivative there. If f has a local extreme at a, then the derivative at a satisfies f ′(a) = 0.

Assume that f has also a second derivative at a.
If f ′(a) = 0 and f ′′(a) < 0, then f has a local maximum at a.
If f ′(a) = 0 and f ′′(a) > 0, then f has a local minimum at a.

Note that the first statement is an implication, not an equivalence. Thus the fact that f ′(a) = 0 does not imply that there is a local extreme at that point. A simple counterexample is the function f (x) = x³ again. For further details see Monotonicity and local extrema in Theory - Graphing functions.

The second statement (classification of extrema) is also just an implication. One can have a local extreme at a and at the same time f ′′(a) = 0. Actually, there exists an advanced version of this theorem that removes this problem.

Theorem (classification of local extrema).
Let a be a point from the domain of a function f, assume that f has all derivatives there and f ′(a) = 0. Let n be the smallest integer for which f ⁽ⁿ⁾(a) is not zero.
If n is odd, then f has a point of inflection at a.
If n is even and f ⁽ⁿ⁾(a) < 0, then f has a local maximum at a.
If n is even and f ⁽ⁿ⁾(a) > 0, then f has a local minimum at a.

This is undoubtedly nice, but it still does not solve the problem of classification, since there can be another catch: What if there is no derivative at a? This is the reason why we usually do the classification not using derivatives but using intervals of monotonicity (see Monotonicity and local extrema in Theory - Graphing functions).

Although we defined monotonicity on general sets, here we reduce our focus on monotonicity on intervals. The reason is simple, the following theorems are (with a few exceptions) all based on MVT where connectedness of the underlying set is crucial. Another reason is that monotonicity is usually investigated on intervals anyway.

Theorem (derivative and global monotonicity).
Let f be a function continuous on an interval I and differentiable on its interior Int(I ).
If f ′ > 0 on Int(I ), then f is increasing on I.
If f ′ ≥ 0 on Int(I ), then f is non-decreasing on I.
If f ′ < 0 on Int(I ), then f is decreasing on I.
If f ′ ≤ 0 on Int(I ), then f is non-increasing on I.
If f ′ = 0 on Int(I ), then f is constant on I.

If f is non-decreasing on I, then f ′ ≥ 0 on Int(I ).
If f is non-increasing on I, then f ′ ≤ 0 on Int(I ).
If f is constant on I, then f ′ = 0 on Int(I ).

Just like the local situation, here we also said everything that was possible in general. In other words, it is not possible to claim that if f is increasing on an interval I, then f ′ should be positive everywhere on Int(I ). And again, f (x) = x³ is the best counter-example, for instance on the interval (−1,1).

Why is connectedness so crucial? Consider the following example:

This function is definitely not constant on the set M, but its derivative is equal to 0 everywhere on M. Similar examples show that none of the statements in the first half of the theorem (where we get monotonicity out of derivative) works in general for non-connected sets. The reason is that these statements are based on MVT, which does not work on non-connected sets (see the section on MVT).

On the other hand, the second half of the theorem, where we deduce things about derivative based on monotonicity, do work for general sets, since these can be proved merely using the definition of derivative.

For practical use of these theorems see Monotonicity and local extrema in Theory - Graphing functions.

We conclude with a corollary that can be quite useful:

Sometimes we just need injectivity on some neigborhood of a given point a. Then one can combine the above theorem and properties of continuity (namely separation from zero, see Continuity in Functions - Theory - Real functions) to get the following statement.

Corollary.
Let f be a function defined on a neighborhood of a. If f ′(a) ≠ 0 and f ′ is continuous at a, then f is monotone and invertible on some neighborhood of a.

Derivative and concavity
Back to Theory - Mean value theorem