3.1 Extrema on an Interval
Consider the function y = sin x over the intervals [0,2!], [0,!/4], and [!,2!].
Use the graph of the function to determine the minimum and maximum.
From the graph of y = sin x for different intervals, it should be clear that the minimum and maximum for a
function will be different over various intervals of the function. Think of “framing up” the section of the graph
that you are interested in when you determine the extrema on a closed interval.
Critical Numbers: Special x values for which the function has a slope of 0, or has an asymptote,
discontinuity or sharp turn. If the function has a maximum or minimum value, it will occur at a
critical number.
Formal Definition:
Let f be defined at c. If f’(c) = 0 or if f’ is undefined at c, then c is a critical number of f.
Example 1: Determine the critical numbers of the function f ( x ) = x 3 ! 3x 2
Page 1
10/4/2010
Strategy to find extrema on a closed interval:
1.
2.
3.
4.
Find the critical numbers of f in (a,b).
Evaluate f at each critical number in (a,b).
Evaluate f at each endpoint of [a,b].
The least of these values is the minimum, the greatest is the maximum.
Example 2: Determine the extrema of f ( x ) = x 2 + 2 x ! 4 on [-5, 1]
Example 3: Determine the extrema of f (x) = 2x ! 3x
3
2
3
on [-1, 3]
9 ! 2.08
Page 2
10/4/2010
Page 3
10/4/2010
Graphics taken from textbook. Calculus of a Single Variable by Larson and Edwards, 9th edition
Page 4
10/4/2010