More about Extrema

Relative Extrema

Relative Maximums

Relative Minimums

• Also called Local Extrema

• Also called Saddle Points

Absolute Extrema

Absolute Maximum

Absolute Minimum

Also called global maximum and global minimum

Critical Points of f

1. ( ) 0f x

A critical point of a function f is a point in the domain of f where

2. ( ) does not existf x(stationary point)

(singular point)

Candidates for Relative Extrema

1. Stationary points:

2. Singular points:

3. Endpoints: endpoints of the domain (if any).

( ) 0.f x

( ) is undefined.f x

Extreme Value TheoremIf a function f is continuous on a closed interval [a, b], then f attains an absolute maximum and minimum on [a, b]. Each extremum must occur at a critical point or an endpoint.

a b a ba b

Attains max. and min.

Attains min. but not max.

No min. and no max.

Interval open Not continuous

Domain Not a Closed IntervalEx. Find the absolute extrema of

1( ) on 3, .2

f xx

Notice that the interval is not closed. Look graphically:

Absolute Max.

(3, 1)

Summary• Finding global maxima and minima is the goal of

optimization. If the function is defined over a closed domain, then by the extreme value theorem global maxima and minima exist. Furthermore, a global maximum (or minimum) must be either a local maximum (or minimum) in the interior of the domain, or it must lie on the boundary of the domain. So a method of finding a global maximum (or minimum) is to look at all the local maxima (or minima) in the interior, and also look at the maxima (or minima) of the points on the boundary; and take the biggest (or smallest) one.