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4.1 Maximum and minimum values
Let f be defined on an interval I containing c. Then
f(c) is the maximum of f on I if f(c)
f(x) for all x in I
In this case, we also say that f has a maximum at c.
f(c) is the minimum of f on I if f(c)
f(x) for all x in I
In this case, we also say that f has a minimum at c.
Also use: absolute maximum, global maximum and absolute minimum, global minimum
The maximum and minimum values are also called extreme values.
EXAMPLES
Extreme Value Theorem If f is continuous on [a, b], then f has
Local (or relative) max and min
If there is an open interval containing c on which f(c) is a maximum, then f(c) is a local maximum.
If there is an open interval containing c on which f(c) is a minimum, then f(c) is a local minimum.
EXAMPLES
Critical numbers
A number c is a critical number (or critical point) of f if either
(1)
or
(2)
4x
x 1
EXAMPLE
Find the critical numbers of f(x) =
Theorem
If f has a local minimum or a local maximum at x = c, then c is a
RECALL: Extreme Value Theorem
and an absolute minimum on [a, b].
2
If f is continuous on [a, b], then f has an absolute maximum
How to find absolute max and absolute min on [a, b]:
1.
2.
Find critical numbers on [a, b]. Say they are c1, c2, …
Compare f(a), f(b), f(c1), f(c2), …
Largest is the max and smallest is the min.
EXAMPLES
1.
f(x) = x3 – 12x on [0, 4]
2.
f(x) = 3x2/3 – 2x on [–1, 1]