Post on 18-Feb-2016
description
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.