1.
2.
3.
4.
Objectives:
Be able to define various vocabulary terms needed to be successful
in this unit.
Be able to understand the definition of extrema of a function on an
interval and “The Extreme Value Theorem”.
Be able to find the relative extrema and critical numbers of a
function.
Be able to find extrema on a closed interval.
Critical Vocabulary:
Extrema, The Extreme Value Theorem, Critical Numbers
I. Vocabulary
Extrema (Plural of Extreme): This means we are talking about ______
(plural of maximum) and _______ (plural
of minimum) of a function.
Interval: Means we are talking about a part of a function,
denoted by interval notation ____ or _____
___________ __________
II. Extrema of a Function
Let f be defined on an interval I containing c
1. f(c) is the _______ of f on I if ____ ≤ ____ for all x in I
2. f(c) is the _______ of f on I if ____ ≥ ____ for all x in I
The _______ and ________ of a function on an interval are
the ________ values, or ______, of the function on the
interval. The minimum and maximum of a function on an
interval are also called the ________ minimum and ________
maximum on the interval.
Example 1: Function: f(x) = x2 + 3, Interval: [-2, 2], Let c = 0
II. Extrema of a Function
If f is _______ on a closed interval [a, b], then f has both
a _________ and ___________ on the interval.
To illustrate this, we will look at the graph of f(x) = x2 + 1
on the following intervals:
III. Relative Extrema and Critical Numbers
Point where a graph changes its behavior (increasing/decreasing)
help in determining the maximum and minimum values of a graph.
Relative Minimum: _____________________________________
Relative Maximum: _____________________________________
III. Relative Extrema and Critical Numbers
1. If there is an open interval containing c on which f(c) is
a maximum, then f (c) is called a _____________ of f.
2. If there is an open interval containing c on which f(c) is
a minimum, then f (c) is called a _____________ of f.
The plural of relative maximum is ___________ and the
plural of relative minimum is _________________.
III. Relative Extrema and Critical Numbers
Example 2: Find the value of the derivative at each of the relative
extrema shown in the graph


9 x2  3
f ( x) 
x3
When the derivative is zero, we call the x-value
associated with it a ____________________.
III. Relative Extrema and Critical Numbers
Example 3: Find any critical numbers algebraically: f(x) = x2(x2 - 4)
1st: Find the derivative
2nd: Set f’(x) = 0
3rd: Check for any places where the derivative is undefined
IV. Finding Extrema on a Closed Interval
To find the extrema of a continuous function f on a closed
interval [a, b], use the following steps:
1. Find the ____________________ of f in (a, b)
2. Evaluate f at each ________________ in (a, b)
3. Evaluate f at each _____________ in [a, b]
4. The least of these values is the ________ and
the greatest is the ______________.
IV. Finding Extrema on a Closed Interval
Example 4: Locate the absolute extrema of the function on the
x2
closed interval
f ( x) 
1st: Find the critical numbers
2nd : Evaluate at the endpoints
x2  3
, [1,1]
IV. Finding Extrema on a Closed Interval
Example 5: Locate the absolute extrema of the function on the
closed interval f ( x)  x1/ 3 , [1,1]
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