Serlo: EN: Derivative and local extrema

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In this chapter we will use derivatives to derive necessary and sufficient criteria for the existence of extrema. In calculus, one often uses the theorem that a function $f:D\to ℝ$ must necessarily satisfy $f^{\prime }(\tilde{x})=0$ so that $f$ has a (local) extremum in $\tilde{x}\in D$. If the derivative function $f^{\prime }$ in addition changes the sign at $\tilde{x}$ , then we have found an extremum. The sign change of the derivative is therefore a sufficient criterion for the extremum. We will now derive this statement and other consequences mathematically and illustrate them with the help of numerous examples. First, however, we will clearly define what an extremum is and what kinds of extrema there are.

Types of extrema

A function $f:D\to ℝ$ can have two types of an extremum: A maximum or a minimum. This in turn can be local or global. As the names already suggest, a local minimum is for example a value $f(\tilde{x})$, which is "locally minimal", i.e. In a neighbourhood of $\tilde{x}$ there is also $f\ge f(\tilde{x})$. Mathematically: There is an interval $(\tilde{x}-ϵ,\tilde{x}+ϵ)$ around $\tilde{x}$, so that $f(x)\ge f(\tilde{x})$ for all arguments $x$ which lie in $(\tilde{x}-ϵ,\tilde{x}+ϵ)$. A global maximum on the other hand is a value $f(\hat{x})$, which is "everywhere maximal". That means, for all arguments $x$ from the domain of definition, we must have $f(x)\le f(\hat{x})$. This intuitive idea is illustrated in the following figure:

Characterization of local and global extrema

For local extrema, a distinction is also made between strict and non-strict extrema. A strict local minimum, for example, is one that is only "strictly" attained at a single point. A non-strict extremum can be attained on an entire subinterval.

Characterization of strict local extrema

We now define the intuitively explained terms formally:

A local maximum/minimum is also sometimes referred to in the literature as relative maximum/minimum, and a strict maximum/minimum as isolated maximum/minimum. With this definition it is also clear that every global extremum is also a local one. Similarly, every strict local extremum is also a local extremum in the usual sense. In the following we want to determine some necessary and sufficient conditions for (strict) local extrema, using the derivative. Unfortunately our criteria are not sufficient to characterise global extrema. Those are a bit harder to catch!

Necessary condition for extrema

Theorem and proof

In order for a function to have a local extremum at a position within its domain of definition, the function must have a horizontal tangent there. This means that the derivative at this point must be zero. This is exactly what the following theorem says:

Application: Intermediate values for derivatives

We have already stated in the previous sections that the derivative function of a differentiable function does not necessarily have to be continuous. An example for this is the following function, which we have learned about in the chapter "derivatives of higher order":

However, it can be shown that the derivative function always fulfils the intermediate value property. The reason why this is not a contradiction is that continuity is a stronger property than the intermediate value property. To prove this, we will use our necessary criterion from the previous theorem. This result is also known in the literature as "Darboux's theorem":

Necessary condition: sign change

Theorem

For many functions it can be tedious to determine only with the necessary condition $f^{\prime }(\tilde{x})=0$ whether $f$ has an extremum in $\tilde{x}$: There is a proof necessary that the function actually does not get greater or lower within the environment. Therefore we are now looking for sufficient conditions for an extremum, which saves us the extra proof work. One possibility is to investigate $f^{\prime }$ in the surroundings of the possible extremum $\tilde{x}$. If the function increases on the left of $\tilde{x}$ and decreases on the right, then there is a maximum. If the function first decreases and then increases, there is a minimum.

Conditions are not necessary

The condition in the previous theorem is a sufficient condition for the existence of an extreme point. There is however no necessary condition. We do not have that an extreme position exists exactly when one of the conditions in the previous sentence is fulfilled. The following example illustrates this.

Necessary condition: presign of the second derivative

Theorem

If $f$ is twice differentiable, we can also use the following sufficient criterion:

Extended sufficient crietrion

The problem with functions like $f(x)=x^4$ is that $f^{″}(0)=12\cdot 0^2=0$ and so the second derivative vanishes. We cannot decide just by the second derivative whether and what kind of extrema are present. If we now differentiate $f$ two more times, we get $f^{(4)}(0)=24>0$. The question now is whether we can conclude from this, analogous to the second criterion, that $f$ in $\tilde{x}=0$ has a strict local minimum.

The answer is "yes" - but there is something we need to take care of: Let us look at the example $g(x)=x^3$. This has, in contrast to $f$ no extremum at $\tilde{x}=0$ , but a saddle point. And this although for the third derivative is also $g^{(3)}(0)=6>0$ . The difference is that here the smallest derivative order, which is not equal to zero, is equal to $3$ and therefore odd. With $f(x)=x^4$ on the other hand, the smallest order is $4$, so it is even. We can generalize this to the following criterion:

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