Are global extrema necessarily local extrema?

Question

Yesterday I asked a question on the existence of local extrema, and have received conflicting comments here. Now from this discussion I'm not sure whether global extrema are necessary local extrema, for a continuous function defined in a metric space. Is this a matter of style / author? or is there a definitive answer?

I would appreciate any references too.

Answer 1

I think the discussion you linked to seemed to conclude that it depended on the author. We can say these things for sure:

• If the global extremum is in the interior of the space (i.e., not on the boundary), it is a local extremum.
• If the global extremum is on the boundary of the space, authors will differ whether or not to call it a local extremum; regardless, many standard methods for identifying local extrema (e.g., taking the derivative) will fail to identify this one.

Answer 2

A global extremum is the largest (smallest) value reached in the interval at hand, it is exists. It is not necessarily a local extremum.

Answer 3

If you want to find careful statements for these types of definitions, I recommend that you look at a Real Analysis text, rather than a Calculus text, as cited in the discussion that you link. If you look, for example, on page 143 of this open text, you'll find:

Definition 4.2.1: Let $S \subset \mathbb R$ be a set and let $f \colon S \to \mathbb R$ be a function. The function $f$ is said to have a relative maximum at $c \in S$ if there exists a $\delta>0$ such that for all $x \in S$ where $|x-c| < \delta$ we have $f(x) \leq f(c)$.

In particular, if $S$ is an interval and we consider $f$ restricted to that interval, then relative extrema can certainly occur at the endpoints.

Answer 4

I suppose it depends on perspective, and on precisely how you choose to define a local extrema. All definitions of local extrema Ive ever seen require all values in the deleted neighborhood to be strictly less than or greater than the point in question. They arent defined in terms of derivatives since not all functions are differentiable, and its a definition that must be established for all optimization problems. The definition holds whether youre talking about an interior point or a boundary point, because youre only considering the domain.

The definition of global minimum is:

For a domain $D\subseteq\Bbb R^n$, the point $x\in D$ is a global minimum of $f:D\to\Bbb R$ if $f(x)\le f(y)$, $\forall y\in D$.

Maximums are similarly defined.

A local minimum is defined slightly differently:

For a domain $D\subseteq\Bbb R^n$, the point $x\in D$ is a local minimum of $f:D\to\Bbb R$ if $\exists U\subseteq D$ such that $f(x)\le f(y)$, $\forall y\in U$.

No mention of derivatives. A local and a global minimum can be the same thing if at a boundary point of the domain. In fact, the local extrema is the global extrema of a sufficiently small subset neighborhood. Of course the neighborhood is a continuous, connected subset of the domain. Lets not get into other messier notions.