Extrema On An Interval

1
Extrema On An Interval

• Section 3.1

2
After this lesson, you should be able to

• Understand the definition of extrema of a
  function on an interval
• Understand the definition of relative extrema of
  a function on an open interval
• Find extrema on a closed interval

3
Extrema
Minimum and maximum values on an interval are
called extremes, or extrema on an interval.

• The minimum value of the function on an interval
  is considered the absolute minimum on the
  interval.

• The maximum value of the function on an interval
  is considered the absolute maximum on the
  interval.

When the just word minimum or maximum is used, we
assume its an absolute min or absolute max.
4
The Extreme Value Theorem
Theorem 3.1 If f is continuous on a closed
interval a, b, then f has both a minimum and a
maximum on the interval.
In other words, if f is continuous on a closed
interval, f must have a min and a max value.
5
Max-Min
f is continuous on a, b
6
Critical Numbers

• c is a critical number for f iff
• f(c) is defined (c is in the domain of f)
• f(c) 0 or f (c) does not exist
• If f has a relative max. or relative min, at
• x c, then c must be a critical number for f.
• The (absolute)max and (absolute)min of f on a,
  b occur either at an endpoint of a, b or at a
  critical number in (a, b).

7
Critical Numbers

• Find all critical s c1, c2, c3cn of f which are
  in
• (a, b).
• Compute f(a), f(b), f(c1), f(c2), f(cn).
• The largest and smallest values in part 2 are the
  max and min of f on a, b.

• To find the max and min of f on a, b

8
Example
Example Find all critical numbers
Domain
9
Example
Example Find all critical numbers.
Domain
10
Example
Example Find all critical numbers.
Domain
11
Example
Example Find the max and min of f on the
interval -3, 1.
Domain
x
f(x)
12
Homework
Section 3.1 page 165 1-23 odd, 55