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Lesson Plan #21
Class: Intuitive Calculus
Date: Friday October 22nd, 2010
Topic: Extrema on an interval
Aim: How do we find extrema on a given interval?
Objectives:
1) Students will be able to find extrema.
HW# 21:
1) Find the value of the derivative (if it exists) at the indicated extremum
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f ( x)  ( x  2) at (2,0)
2) Find any critical numbers of the function f ( x)  x 2 ( x  3)
3) Determine the absolute extrema of the function f ( x)  x 3  3x 2 in the interval  1,3
Do Now:
Sketch the graph of f ( x)  x 2  1in the interval [1,2]
Procedure:
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
Example:
Find the minimum and maximum values of the function
f ( x)  x 2  1in the
interval [1,2]
Example:
Find the minimum and maximum values of the function
interval
 1,2.
f ( x)  x 2  1 in the
Definition of Extrema: Let f be defined on an interval I containing c .
1) f (c ) is the minimum of f in the interval if f (c)  f ( x) for all
in the interval I.
x
2) f (c ) is the maximum of f in the interval if f (c)  f ( x) for all
x in the interval I.
The minimum and maximum values of a function on an
interval are called the extreme values, or extrema, of the function in the
interval. When you consider extrema within a closed interval,
the minimum and maximum of a function are also called absolute
minimum and absolute maximum (or global minimum and global
maximum.
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Here is a related theorem, called the Extreme Value Theorem.
The Extreme Value Theorem: If f is continuous on a closed
the interval.
interval, then f
has both a minimum and maximum in
We see that extrema can occur at interior points or endpoints of the interval. Extrema that occur at the endpoints are called
endpoint extrema
Determine from the graph whether or not f possesses a minimum on the given interval (a,b)
A)
B)
Does the function at right have a minimum in the interval
 2,4 ?
Does the function at right have a maximum in the interval
 2,4 ?
For this function, is there an open interval for which there is a maximum or minimum?
Definition of Relative Extrema:
If there is an open interval in which f (c ) is a maximum, then f (c ) is called a relative
maximum of f . (Also called local maximum)
If there is an open interval in which f (c ) is a minimum, then f (c ) is called a relative
minimum of f . (Also called local minimum)
Example:
Find the value of the derivative at each of the relative extrema shown in the graph of
f ( x)  x 3  3 x .
Note that relative extrema occur where the derivative is zero or where the derivative
is undefined.
Definition of a critical number: If f is defined at c , then
critical number of f if f ' (c )  0 or if f ' is undefined at
c is called a
c
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Example #1:
Find any critical numbers of the function
A) 𝑔(𝑥) = 𝑥 2 (𝑥 − 3)
B) g (t )  t 4  t
Theorem: Relative Extrema Occur Only At Critical Numbers:
If f has a relative maximum or relative minimum at x  c , then
c is a
critical number of f
Example: Find the absolute extrema (not necessarily relative since it is closed interval) of
f ( x)  3x 4  4 x 3 on the interval
1, 2 (Hint: Start by testing endpoints in the function and then critical numbers; the low value is the minimum and the high
value is the maximum)
Example:
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Find the extrema of
interval)
f ( x)  2 x  3x 3 on the interval  1,3 . (This would indicate absolute extrema since it is on an open