MAXIMUM and MINIMUM VALUES
(Extreme Values on an Interval)
The maximum and minimum values of a function on an interval are also
called extreme values or absolute extrema.
Extreme values of a function f(x) on a given interval [a,b] may occur at:
 a peak (max) or valley (min) where f ‘(x) = 0
(a horizontal tangent exists)
 an end point on the given interval (at x = a or x = b)
 a point where f ‘(x) DNE (a cusp or corner exists)
For example, consider the following cases:
(Note: If f ‘(a) = 0 does this necessarily indicate an extreme value at x = a?)
Algorithm for Determining Extreme Values:
For a function, f(x), that is continuous at every point on an interval, [a,b]:
1. Determine all points in [a,b] where f ‘(x) = 0 or f ‘(x) DNE
(these are called critical numbers).
2. Evaluate f(x) at the endpoints, a and b, and at the critical numbers.
3. Compare all the values found in step 2.
 the largest of these values is the maximum
 the smallest of these values is the minimum
Example 
Determine the extreme values of f(x) = x4 + 4x3 – 2x2 – 12x + 9, x  [–2,2].
y
x
Example 
State why the algorithm can or cannot be used to determine the maximum
or minimum values of each of the following:
a) y  5 x , –1 ≤ x ≤ 2
x 1
2
b) y  x  4 , x  [–1,3]
x 3
Example 
Technicians working for the Ministry of Natural Resources have found that
the amount of a pollutant in a river can be represented by the given
function, P(t), where t is the time in years since a clean–up campaign
started. If the clean–up campaign started on January 1st, determine the day
on which the pollutant was at its lowest level.
P( t )  2t 
Homework:
1
,0≤t≤1
162t  1
p.135–138 #1, 2, 3ace, 4, 5a, 6, 7a, 8, 11–13
2
3
Also: Determine the maximum and minimum values of the function 𝑓 𝑥 = (𝑥 − 9) .
2