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Extrema On An Interval

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Understand the definition of extrema of a function on an interval ... f(c) is defined (c is in the domain of f) f '(c) = 0 or f '(c) does not exist ... – PowerPoint PPT presentation

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Title: 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.

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.
5
Examples
  • Given , name any
    extrema of f on the interval
  • 0, 5
  • (0, 5)

6
Relative Extrema
  • Relative extrema are turning points of the graph.
  • The turning points may occur as smooth hills or
    valleys.
  • The turning points may occur as sharp turns.

In these cases, the function is not
differentiable at the relative max/min.
7
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).

8
Finding Extrema on a Closed Interval
To find the max and min of f on a, b
  • Find all critical s of f which are in (a, b).
  • Find all values of x for which f (c) 0 or f
    (c) does not exist
  • Evaluate f at each of the critical values
  • Plug each of the critical values into the
    function to find the y-coordinate.
  • Evaluate f at each endpoint
  • Find f(a) and f(b)
  • The smallest value from parts 2 3 is the
    minimum and the largest value from parts 2 3 is
    the maximum of f on a, b.

9
Example
Example Find all critical numbers
Domain
10
Example
Example Find all critical numbers.
Domain
11
Example
Example Find all critical numbers.
Domain
12
Example
Example Find the max and min of f on the
interval 0, 4.
Domain
13
Example
Example Find the max and min of f on the
interval -1, 1.
Domain
14
Example
Example Graph a function f on the interval -3,
4 that has the given characteristics.
  • Relative max at x -2
  • Absolute min at x 2
  • Absolute max at x 4

15
Homework
Section 3.1 page 169 1-25 odd, 37, 45
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