Extreme Values

Local Maximum

Local Minimum

In the first page of this Lesson (The Shape of a Function) we discussed local maxima and local minima as points where a continuous function "turns around".

We are now interested in a more precise definition as well as a working notion of absolute maxima and minima. This is easy to define:

Absolute Extrema

Let f(x) be defined on a set S.

Then f(x) has an absolute maximum (minimum) at a point s in S if f(s) f(x) (f(s) f(x)) for all x in S.

The Extreme Value Theorem guarantees that a continuous function attain both an absolute maximum and absolute minimum on a closed interval. It does NOT guarantee that there be a unique absolute maximum or minimum, nor does it say that absolute extrema must occur in the interior of a closed interval. The following graphs indicate what kinds of things can happen. The last two graphs are of discontinuous functions. In each case the absolute maxima are indicated as green points on the graph. The domains of all graphs are CLOSED intervals.

Unique maximum.

Two absolute maxima.

Maximum at Endpoint.

No Maximum.

Isolated Maximum.

To find absolute extreme it is helpful to locate relative or local extrema. The definition is:

Local (or Relative) Extrema

Let f(x) be defined on a set S.

Then f(x) has a local maximum (minimum) at a point s in S if there is positive distance d such that f(s) f(x) (f(s) f(x)) for all x in S with |x - s| < d.

Absolute maxima and minima are also local maxima and minima, but local extreme points do not have to be absolute extrema.

Here are some graphs illustrating local minima. The local minima which ARE NOT absolute minima are indicated in blue while those which ARE absolute minima are indicated in green

Three Local Minima.

Three Local Minima.

One Local Minimum.

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