5.1: Extreme Values of Functions
Definition: Absolute Extreme Values
Let 𝑓 be a function with domain 𝐷. Then 𝑓(𝑐) is the:
(a) Absolute maximum value on 𝐷 if and only if 𝑓(𝑥) ≤ 𝑓(𝑐) for all 𝑥 in 𝐷.
(b) Absolute minimum value on 𝐷 if and only if 𝑓(𝑥) ≥ 𝑓(𝑐) for all x in 𝐷.
**Absolute or global max and min values are also called absolute extrema
Example 1: For the function 𝑦 = 𝑥 2 , find all absolute extrema on the following domains:
(a) (−∞, ∞)
(b) [0,2]
(c) (0, 2]
(d) (0, 2)
Extreme Value Theorem: If 𝑓 is continuous on a closed interval [𝑎, 𝑏], then 𝑓 has both a maximum and
minimum value on the interval. (Draw a sketch below)
Definition: Local Extreme Values
Let 𝑐 be an interior point of the domain of 𝑓. Then 𝑓(𝑐) is a:
(a) Local maximum value at 𝑐 if and only if 𝑓(𝑥) ≤ 𝑓(𝑐) for all 𝑥 in some open interval containing
𝑐.
(b) Local minimum value at 𝑐 if and only if 𝑓(𝑥) ≥ 𝑓(𝑐) for all 𝑥 in some open interval containing 𝑐.
*𝑓 has a local max or local min at an endpoint c if the appropriate inequality holds for some half-open
domain interval containing 𝑐.
**local extrema are also called relative extrema
**an absolute extremum is also a local extremum
The figure below summarizes how we classify extreme values:
**************Finding Extreme Values**************************
Local Extreme Values: If 𝑓 has a local max or local min at an interior point 𝑐 of its domain, and if 𝑓′
exists at 𝑐, then 𝑓 ′ (𝑐) = 0.
Definition: Critical Point
A point on the interior of the domain 𝑓 at which 𝑓 ′ = 0 or 𝑓′ DNE is a critical point of f.
Definition: Stationary Point
A point on the interior of the domain 𝑓 at which 𝑓 ′ = 0 is a stationary point of f.
** We only need to look for extrema where 𝑓 ′ = 0, where 𝑓′ DNE, or at the endpoints of a given closed
interval. This is known as the Candidate’s Test.
Example 2: Find the extreme values of the function and where they occur.
(a) 𝑓(𝑥) = 3𝑥 5 − 5𝑥 3 − 1 on [−2, 2]
(b) 𝑓(𝑥) =
ln 𝑥
𝑥
on [1, 3]
5.1: Extreme Values of Functions
Definition: Absolute Extreme Values
Let 𝑓 be a function with domain 𝐷. Then 𝑓(𝑐) is the:
(a) Absolute maximum value
(b) Absolute minimum value
on 𝐷 if and only if 𝑓(𝑥) ≤ 𝑓(𝑐) for all 𝑥 in 𝐷.
on 𝐷 if and only if 𝑓(𝑥) ≥ 𝑓(𝑐) for all x in 𝐷.
**Absolute or global max and min values are also called absolute extrema
Example 1: For the function 𝑦 = 𝑥 2 , find all absolute extrema on the following domains:
(a) (−∞, ∞)
(b) [0,2]
(c) (0, 2]
(d) (0, 2)
Extreme Value Theorem:
:
If 𝑓 is continuous on a closed interval [𝑎, 𝑏], then 𝑓 has both a
maximum and minimum value on the interval. (Draw a sketch below)
Definition: Local Extreme Values
Let 𝑐 be an interior point of the domain of 𝑓. Then 𝑓(𝑐) is a:
(a) Local maximum value at 𝑐 if and only if 𝑓(𝑥) ≤ 𝑓(𝑐) for all 𝑥 in some open interval containing
𝑐.
(b) Local minimum value at 𝑐 if and only if 𝑓(𝑥) ≥ 𝑓(𝑐) for all 𝑥 in some open interval containing 𝑐.
*𝑓 has a local max or local min at an endpoint c if the appropriate inequality holds for some half-open
domain interval containing 𝑐.
**local extrema are also called relative extrema
**an absolute extremum is also a local extremum
The figure below summarizes how we classify extreme values:
**************Finding Extreme Values**************************
Local Extreme Values: If 𝑓 has a local max or local min at an interior point 𝑐 of its domain, and if 𝑓′
exists at 𝑐, then 𝑓 ′ (𝑐) = 0.
Definition: Critical Point
A point on the interior of the domain 𝑓 at which 𝑓 ′ = 0 or 𝑓′ DNE
is a critical point of f.
Definition: Stationary Point
A point on the interior of the domain 𝑓 at which 𝑓 ′ = 0 is a stationary point of f.
** We only need to look for extrema where 𝑓 ′ = 0, where 𝑓′ DNE, or at the endpoints of a given closed
interval. This is known as the Candidate’s Test.
Example 2: Find the extreme values of each function and where they occur.
(a) 𝑓(𝑥) = 3𝑥 5 − 5𝑥 3 − 1 on [−2, 2]
(b) 𝑓(𝑥) =
ln 𝑥
𝑥
on [1, 3]