Post on 17-Jan-2016
description
Behavior of Functions and
Their Graphs,
ROMMEL O. GREGORIO
OUTLINE
Maximum and minimum values
Increasing and decreasing functions and the first-
derivative test
Concavity, points of inflection, and the second-
derivative test
Sketching graphs of functions and their
derivatives
Summary of sketching graphs of functions
Optimization Problems
MAXIMUM AND MINIMUM FUNCTION
VALUES
MAXIMUM AND MINIMUM FUNCTION
VALUES
Definition of a Relative Maximum Value
The function f has a relative maximum value at
the number c if there exists an open interval
containing c, on which f is defined, such that
f(c)≥f(x) for all x in the interval.
Definition of a Relative Minimum Value
The function f has a relative minimum value at
the number c if there exists an open interval
containing c, on which f is defined, such that
f(c)≤f(x) for all x in the interval.
MAXIMUM AND MINIMUM FUNCTION
VALUES Definition of an Absolute Maximum Value
The function f has an absolute maximum value on an
interval if there is some number c in the interval such
that f(c)≥f(x) for all x in the interval. The number f(c)
is then the absolute maximum value of f on the
interval.
Definition of an Absolute Minimum Value
The function f has an absolute minimum value on an
interval if there is some number c in the interval such
that f(c)≤f(x) for all x in the interval. The number f(c)
is then the absolute minimum value of f on the
interval.
MAXIMUM AND MINIMUM FUNCTION
VALUES
Theorem
If f(x) exists for all values of x in the open interval
(a, b), and if f has a relative extremum at c,
where a<c<b, and if f’(c) exists, then f’(c)=0
If f has a relative extremum at c, and if f’(c)
exists, then the graph of f must have a horizontal
tangent line at the point where x=c.
MAXIMUM AND MINIMUM FUNCTION
VALUES
Illustration
f(x)=x2-4x+5
x
y
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VALUES
QUESTION: If f’(c)=0, does
it mean that we have a
relative extremum at c?
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VALUES
Consider
f(x)=(x-1)3+2
x
y
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VALUES
QUESTION: If f’(c) does not
exist, is it possible for f to
have a relative extremum
at c?
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VALUES
Consider
𝑓 𝑥 = 2𝑥 − 1 𝑖𝑓 𝑥 ≤ 38 − 𝑥 𝑖𝑓 𝑥 > 3
x
y
MAXIMUM AND MINIMUM FUNCTION
VALUES
Remarks:
1. f’(c) can be equal to zero even if f does not have
a relative extremum at c.
2. f may have a relative extremum at a number at
which the derivative fails to exist.
3. it is possible that a function f can be defined at a
number c where f’(c) does not exist and yet f may
not have a relative extremum there.
MAXIMUM AND MINIMUM FUNCTION
VALUES
In summary, if a function f is defined
at a number c, a necessary condition
for f to have a relative extremum there
is that either f’(c)=0 or f’(c) does not
exist. But this condition is not
sufficient.
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Definition of a Critical Number
If c is a number on the domain of the function f,
and if either f’(c)=0 or f’(c) does not exist, then c
is a critical number of f.
Examples
1. f x = x4 + 4x3 − 2x2 − 12x
2. f x =x2+4
x−2
3. f x = x − 2
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Example 1
f(x)=4-3x on (-1, 4]
x
y
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VALUES
Example 2
𝑓 𝑥 = 4 − 𝑥2; (-2, 2)
x
y
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VALUES
Example 3
𝑔 𝑥 = 𝑥 + 1 𝑖𝑓 𝑥 ≠ −13 𝑖𝑓 𝑥 = −1
; [-2, 1]
x
y
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Example 4
ℎ 𝑥 =4
(𝑥−3)2; [2, 5)
x
y
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VALUES
The Extreme Value Theorem
If the function f is continuous on
the closed interval [a, b], then f
has an absolute maximum value
and an absolute minimum value on
[a, b].
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VALUES
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Steps in Finding the absolute Extremum of f on [a,
b]
1. Find the function values at the critical numbers of
f on (a, b)
2. Find the values of f(a) and f(b)
3. The largest of the values from steps 1 and 2 is the
absolute maximum value, and the smallest of the
values is the absolute minimum value
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VALUES
Examples. Find the absolute extrema of the
following functions on the given interval.
1. f x = x3 + 5x − 4, [-3, -1]
2. g x =x
x+2, [-1, 2]
3. h x = x4 − 8x2 + 16, [-1, 2]
4. F x = x3 − 5x − 4, [-3, -1]