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MATH 1910 - Extreme Values
 Absolute Extrema.
A function f has an absolute maximum (or global maximum) at x  c if f c ≥ f x for all x in the
domain of f. In this case, f c is called the maximum value of f over its domain.
A function f has an absolute minimum (or global minimum) at x  c if f c ≤ f x for all x in the
domain of f. In this case, f c is called the minimum value of f over its domain.
The maximum and minimum values of f are called the extreme values or extrema of f (singular:
extremum).
A function may NOT have a maximum or minimum value. For example, the function below has a
minimum value of −1 at x  1, but it has NO maximum value.
y
10
8
6
4
2
-4
-2
2
4
6
x
-2
 Relative Extrema. Here we consider maximum and minimum values over a small interval of the
function’s domain.
A function f has a local maximum (or relative maximum) at x  c if fc ≥ f x for all x near c (i.e.
on some open interval about c). In this case, f c is the local maximum (plural: maxima).
A function f has a local minimum (or relative minimum) at x  c if fc ≤ f x for all x near c (i.e.
on some open interval about c). In this case, f c is the local minimum (plural minima).
For example, the function below has local maxima of about 1. 8 at x ≈ −1. 25 and 0 at x ≈ 3. 2. It also
has local minima of about 0 at x ≈ −3. 1 and about −1. 8 at x ≈ 1. 25. Note that neither local maxima
is an absolute maximum, nor is either local minima an absolute minimum.
4
2
-5
-4
-3
-2
-1
1
2
3
4
5
-2
-4
 The Extreme Value Theorem.
If f is continuous on a closed interval a, b, then f has an absolute maximum f c and an absolute
minimum of f d for c and d both in the closed interval a, b.
Note that the extreme value may occur more than once in the interval a, b.
 Fermat’s Theorem.
If f has a local maximum or minimum at x  c, and f ′ c exists, then f ′ c  0.
Note that the inverse of this is NOT true. If f ′ c  0, that does NOT necessarily mean that f has a
local maximum or minimum at x  c (See Figure 9 on page 272 of your text).
There are cases where f can have a local maximum or minimum at x  c even though f ′ c does not
exist. An example is the function f x  |x|. It has a local (and absolute) minimum at x  0, but f ′ 0
does not exist.
 Critical Number.
A critical number of a function f is a number c in the domain of f such that either f ′ c  0 or f ′ c
does not exist.
Find the critical numbers in each case:
1. px  x 3 − 3x  1
2. f x  x 3/5 4 − x
3. gt  t − 2 cos t
4. hx  xe 2x
 Fermat’s Theorem Rephrased:
If f has a local maximum or minimum at x  c, then c is a critical number of f.
Note that the inverse of this is not true. Thus, if c is a critical number of f, then f does NOT
necessarily have a local maximum or minimum at x  c.
 Finding extreme values on a closed interval.
Find the extreme values of f on the given interval.
1. px  x 3 − 3x  1, 0, 3
2. f x  x 3/5 4 − x, −32, 4
3. gt  t − 2 cos t, −, 
4. hx  xe 2x , −1, 0
 Rolle’s Theorem.
If f x is continuous on a, b and differentiable on a, b with f a  f b, then there exists a
number c in the interval a, b such that f ′ c  0.