Name: __________________________________ Period ______ Date ______________ Sec3-1 Extrema on Intervals
Day1: Sec3.1 #1-15 all, 54, 57-60, 67, 68 all
Day 2: #17-35 odd, 37-40 all
Maximum or minimum y-values on intervals, are called EXTREMA.
There are two kinds of extreme y-values: Absolute and Relative.
(1) ABSOLUTE extrema on an interval is the largest or smallest y-value in that interval.
Absolute Extrema are also called the GLOBAL maximum or GLOBAL minimum.
Extreme Value Theorem:
A function that is continuous on a closed interval always has a maximum and a minimum on that interval.
Closed Interval [-2, 3]
Open Interval (-2, 3)
Has both an absolute
maximum and an absolute
minimum
Does not always have an
absolute maximum or
an absolute minimum
(2) RELATIVE maximum or minimum y-values exist on open intervals are also LOCAL maximum or minimum.
If there is an open interval that has a maximum or a minimum y-value, then those y-values are relative extrema.
Example #1: On the CLOSED interval, [-2, 3]
Identify any extrema on the CLOSED interval
as absolute and/or relative
Example #2: On the OPEN Interval, (-2, 3)
Identify any extrema on the CLOSED interval
as absolute and/or relative
9. Identify any extrema on the CLOSED
interval as absolute and/or relative.
10. Identify any extrema on the CLOSED
interval as absolute and/or relative.
Explain why one of these graphs does NOT have an absolute minimum.
How can we use Calculus to find these extreme values from an equation, rather than from a graph.
(a)
(b)
(d)
(e)
Consider the following examples of
relative extreme values
(maximums and minimums).
What is true about the derivative
at all these extreme values?
(c)
What do you notice about the value of the derivative at relative minimum values and relative maximum values?
CRITICAL NUMBERS:
Let f be defined at c . If f (c)  0 or if f is not differentiable at c , then c is a critical number of f .
Theorem: If f has a relative minimum or maximum at x  c , then c is a critical number of f .
It is important to note that if you have a critical value you MAY have a relative extreme y-value.
But, if you have a relative extreme y-value then it definitely has a critical value associated with it.
Examples: Find any critical numbers for the following functions. Then graph to check location of any extrema.
You need to take the derivative and look for values of x that make the derivative zero or undefined.
(a) f ( x)  3x 2  6 x  1
Examples: Find any critical numbers for the following functions. Then graph to check location of any extrema.
(b) f ( x)  x 5  x
(c) f ( x)  cos 2 ( x)  sin( x) Find all critical values on the interval: 0  x  2
Day 2
Extreme y-values on closed intervals can occur at the endpoints or in the open interval.
Use critical values to find the extrema on the open interval.
Then check that the y-values at each endpoint to see if they are the absolute max or min y-values.
Examples: Find the extrema on a closed intervals
(1) Find the critical number of f on the open interval (a,b)
(2) Evaluate f at each of the critical numbers of f in the open interval (a,b)
(3) Evaluate f at each endpoint: x = a and at x = b
(4) The least is the minimum and the greatest is the maximum.
(a) f ( x)  x 4  10 x 2 on the interval [ 1, 4]
Find the critical values when f  is zero or undefined.
Evaluate f ( x)  x 4  10 x 2 at all the critical values and at the endpoints of the interval: x  1 & x  4
At the endpoint
At the critical values
At the critical value
At the endpoint
y  f (1)
y f(
)
y f(
)
y  f (4)
Sketch the graph to check the location of the extrema on this closed interval.
(b) f ( x)  ( x  3)
4
5
on the interval [ 1, 4] . Find the critical values when f  is zero or undefined.
4
Evaluate f ( x)  ( x  3) 5 at all the critical values and at the endpoints of the interval: x  1 & x  4
At the endpoint
At the critical value
At the endpoint
y  f (1)
y f(
)
y  f (4)
Sketch the graph to check the location of the extrema on this closed interval.
(c) f ( x)  cos2 x  sin x on the interval [0, 2 ] . Find the critical values when f  is zero or undefined.
Evaluate f ( x)  cos2 x  sin x at all the critical values and at the endpoints of the interval: x  0 & x  2
At the endpoint
At the critical values
At the critical value
At the endpoint
y  f (0)
y f(
)
y f(
)
y  f (2 )
Sketch the graph to check the location of the extrema on this closed interval.
(d) f ( x)  5  x  5 on the interval [1,10] Sketch the graph first and then evaluate the derivative.
Find the critical values when f  is zero or undefined.
(e)
f ( x)  [[ x]] on the interval [4,8] \
Sketch the graph first and then evaluate the derivative.
Use y is the greatest integer less than or equal to the input value, x
Find the critical values when f  is zero or undefined.
Table
X
0
0.2
0.5
0.8
1
1.5
2
2.6
3
Y
0
1
2
3
2
(f) Locate the absolute extrema of the function, f ( x)  25  x , (if any exists) over each interval.
2
First, sketch the graph of f ( x)  25  x
(a) [ 5,5]
(b) (5,5)
(c) [5, 0)
(d) [2,5)