Calc 1 Lecture Notes
Section 3.3
Page 1 of 4
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 4
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.
Strategy for finding extrema on a closed interval:
1. Find all critical numbers in the closed interval, and compute function values at these
points.
2. Compute function values at the endpoints.
3. Use these values to identify the absolute maximum and absolute minimum.
Practice:
1. Find all critical numbers and extrema of f  x   x4  6x2  2 .
Calc 1 Lecture Notes
Section 3.3
2. Find all critical numbers and extrema of f  x    2 x  5
3. Find all critical numbers and extrema of f  x   x3 .
4. Find all critical numbers and extrema of f  x   5 x .
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 3  2 x  5 .
2
Calc 1 Lecture Notes
Section 3.3
Page 4 of 4
2
1
 52

5. Find all critical numbers and extrema of f  x    x  x 5  .


y

y = (root(5,x^2)
- root(5,x))^2






6. Find all absolute extrema of f  x   xe
cos 2 x 



on [-2, 2] and [2, 5].






x
