Download Section 4.1 Maximum and Minimum Values

4.1
Maximum and Minimum Values
Maximum Values
Absolute Maximum
Local Maximum
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c1
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Minimum Values
Absolute Minimum
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c2
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Collectively, maximum
and minimum values are
called extreme values.
Local Minimum
Definitions
A function f has an absolute (global) maximum at c
if f (c) ≥ f (x) for all x in the domain.
A function f has an absolute (global) minimum at c if
f (c) ≤ f (x) for all x in the domain.
The maximum and minimum values are called
extreme values.
A function f has a local (relative) maximum at c if
f (c) ≥ f (x) where x is in a small open interval about c.
A function f has a local (relative) minimum at c if
f (c) ≤ f (x) where x is in a small open interval about c.
The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f
attains an absolute maximum value f (c) and an
absolute minimum f (d) at some numbers c and d
in [a, b].
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Fermat’s Theorem
If f has a local maximum or minimum at c, and
if f '(c) exists, then f '(c) = 0.
Question If a (local) maximum or minimum occur at c,
then what is the value of f '(c)?
Critical Number or Value
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 ( f (x) is not differentiable)
Fact
An absolute extremum occurs at two places:
 Critical points
 End points
Finding Absolute Extrema on a
Closed Interval [a,b]
1. Find the critical numbers of f on (a, b).
2. Compute the value of f at each of
• the critical numbers on (a,b)
• the endpoints a and b.
3. The largest of these values is the absolute maximum.
The smallest is the absolute minimum.
Examples
Locate the absolute extrema of the function on
the closed interval.
f(x) = x3 – 12x
on
g(x) = 4x / (x2+1)
[-3,4]
on [0,3]
h(t) = 2 sec(t) - tan(t)
on [0,π/4]
f (t )  t  4t  4t  1
4
8
3
3
2
3
2
on [0, 2]
h( x)  x  16 x  3 on [3,1]