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Calc 1 Lecture Notes
Section 3.3
Page 1 of 2
Section 3.3: Maximum and Minimum Values
Big idea: Derivatives can be used to compute properties of the graph of a function: the location
of an extremum of a function is where the derivative is equal to zero, where the derivative is
undefined, or, if the function is defined on a closed interval, at the endpoints of that closed
interval.
Big skill: You should be able to find the extrema of a function by calculating where the
derivative is zero or undefined, and by evaluating the function at its endpoints (if it is defined on
a closed interval).
Definition 3.1: Absolute Extrema
If a function f is defined on a set of numbers S, then for a number c  S:
i.
f(c) is the absolute maximum of f on S if f(c)  f(x) for all x  S.
ii.
f(c) is the absolute minimum of f on S if f(c)  f(x) for all x  S.
An absolute extremum is either an absolute maximum or an absolute minimum. The plural
form of the word extremum is extrema.
Theorem 3.1: Extreme Value Theorem
A continuous function f on a closed, bounded interval [a, b] attains both an absolute maximum
and an absolute minimum on that interval.
Definition 3.2: Local Extrema
If a function f is defined on a set of numbers S, then for a number c  S:
i.
f(c) is a local maximum of f on S if f(c)  f(x) for all x in some open interval
containing c.
ii.
f(c) is a local minimum of f on S if f(c)  f(x) for all x in some open interval
containing c.
A local extremum is either a local maximum or a local minimum.
Definition 3.3: Critical Number
A number c in the domain of a function f is called a critical number of f if f (c) = 0 or f (c) is
undefined.
y


x













 x3  x

 59  x  5   45

 16 
4  64

9
f  x    227 
x   1
 32 
4

x  1.25


2
  x  4   0.75
x  1.25
1.25  x  1.75
1.75  x  2.25
2.25  x  3
x3
Calc 1 Lecture Notes
Section 3.3
Page 2 of 2
Theorem 3.2: Fermat’s Theorem
If f(c) is a local extremum, then c must be a critical number of f.
Note: all extrema occur at critical values, but not all critical values are extrema.
Theorem 3.3: Absolute Extrema and Closed Intervals
If f is continuous on the closed interval [a, b], then the absolute extrema of f must occur at an
endpoint or at a critical number.
Practice:
1. Find all critical numbers of f  x   x4  6 x2  2
1
 2

2. Find all critical numbers of f  x    x 5  x 5 


3. Find all absolute extrema of f  x   xe
cos 2 x 
2
on [-2, 2] and [2, 5]