Title: Extrema On An Interval
1Extrema On An Interval
2After 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
3Extrema
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.
4The 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.
5Examples
- Given , name any
extrema of f on the interval - 0, 5
- (0, 5)
6Relative 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.
7Critical 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).
8Finding 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.
9Example
Example Find all critical numbers
Domain
10Example
Example Find all critical numbers.
Domain
11Example
Example Find all critical numbers.
Domain
12Example
Example Find the max and min of f on the
interval 0, 4.
Domain
13Example
Example Find the max and min of f on the
interval -1, 1.
Domain
14Example
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
15Homework
Section 3.1 page 169 1-25 odd, 37, 45