5'2 Relative and Absolute Extreme Points

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5.2 Relative and Absolute Extreme Points
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Increasing and Decreasing
How can we determine the intervals of x (from
left to right) for which f (output values) is
increasing or decreasing?
m lt 0
m gt 0
Decreasing
Increasing
Increasing
m gt 0
b
a
(a) If f '(x) gt 0 on an interval, then f is
increasing on that interval.
(b) If f '(x) lt 0 on an interval, then f is
decreasing on that interval.
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Relative Extreme Values
We can use tangent lines to identify relative
(local) maximums and minimums.
f (a) is a Relative Maximum
It is important to distinguish where an extreme
value occurs from what they are. For example, f
has a relative maximum at x a, but the relative
maximum is f (a).
m 0
m lt 0
m gt 0
m gt 0
b
a
m 0
f (b) is a Relative Minimum
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Finding Relative Extreme Values
Obviously, if there is a relative maximum or
minimum, it must occur at points where f '(x)
0. What other conditions must be true to have a
relative extreme value? If f '(x) 0, does this
guarantee relative extreme value?
f (a) is a Relative Maximum
m 0
m gt 0
m 0
m lt 0
m gt 0
m gt 0
b
a
a
m gt 0
m 0
f (b) is a Relative Minimum
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Finding Relative Extreme Values
Relative extreme values can occur at cusp points
too. Here f '(x) undefined since the slope of
the tangent line cannot be determined. Look for
these as well.
f (a) is a Relative Maximum
a
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Finding Relative Extreme Values
Suppose that f '(c) 0 or undefined. Then x c
is called a critical number of f.
(a) If f '(x) changes from positive to negative
at c, then f has a local maximum at c. The graph
of f '(x) will cross (not just touch) the input
axis and change from positive to negative.
(b) If f '(x) changes from negative to positive
at c, then f has a local maximum at c. The graph
of f '(x) will cross (not just touch) the input
axis and change from negative to positive.
(c) If f '(x) changes does not change sign at c,
then f has a no local extreme values at c. The
graph of f '(x) will touch, but not cross the
input axis and will not change signs.
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Finding Absolute Extreme Values
f (b) and f (d) are absolute maximums
(a) If f (c) f (x) for all values of x in the
domain of f (or on a closed interval a, e),
then x c is called an absolute maximum.
Although there can be only one absolute maximum,
it can occur at multiple values of x.
a
c
e
b
d
f (c) is an absolute minimum
(b) If f (c) f (x) for all values of x in the
domain of f (or on a closed interval a, e),
then x c is called an absolute minimum.
Although there can be only one absolute minimum,
it can occur at multiple values of x.
f (e) is an absolute maximum
a
c
e
b
Caution Always compare the function values at
the critical numbers to the function values at
the end points of closed intervals when looking
for absolute extreme values.
f (a) is an absolute minimum
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Examples
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