Download Absolute Extreme Values

AP Calculus BC
Student Notes – Chapter 4
Absolute Extreme Values
f
If
is a function with domain D, then
f (c) is the:
1.
Absolute maximum value iff
f (x)  f (c)
given any
x D .
2.
Absolute minimum value iff
f (x)  f (c)
given any
x D .
Extreme Value Theorem
If f is continuous on a closed interval [a, b], then
minimum on the interval.
f
has both a maximum and
Local Extrema
If
c
is an interior point of
f
, then
f (c) is:
1. A local maximum at
containing c .
c
iff
f (x)  f (c)
given any
x
in an open interval
2. A local minimum at
containing c .
c
iff
f (x)  f (c)
given any
x
in an open interval
Theorem
If
f
has any local extrema at an interior point
c
of its domain, and if
f (c)  0 .
then
f  exists at c ,
Definition
Let
f
be defined at
critical point of
f
c.
If
f (c)  0 or if f
is not differentiable at
c , then c
is a
.
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AP Calculus BC
Student Notes – Chapter 4
Rolle’s Theorem
Let f be continuous on the closed interval [a, b] and differentiable on the open interval
(a, b). If
f (a)  f (b)
then there is at least one number
c
in (a, b) such that
f (c)  0.
Mean Value Theorem
If f is continuous on the closed interval [a, b] and differentiable on the open interval (a,
b), then there exists a number c such that
f (c) 
f (b)  f (a)
ba
Definition of Increasing and Decreasing Functions
f is increasing on an interval if for any two numbers x1 and x2
x1  x2 implies f ( x1)  f ( x2 ).
in the
f is decreasing on an interval if for any two numbers x1 and x2
x1  x2 implies f ( x1)  f ( x2 ).
in the
A function
interval,
A function
interval,
Test for Increasing and Decreasing Functions
Let f be a function that is continuous on the closed interval [a, b] and differentiable on
the open interval (a, b).
1. If
f ( x)  0 for all x
in (a, b), then
f
is increasing on [a, b].
2. If
f ( x)  0 for all x
in (a, b), then
f
is decreasing on [a, b].
3. If
f ( x)  0 for all x
in (a, b), then
f
is constant on [a, b].
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AP Calculus BC
Student Notes – Chapter 4
First Derivative Test for Local Extrema
If f is a continuous function on an open interval I with a critical number c in I and if
differentiable on I, except possibly at c, then:
f
is
1.
If
f ( x) changes sign from negative to positive at c, then f (c) is a relative minimum.
2.
If
f ( x) changes sign from positive to negative at c, then f (c) is a relative maximum.
3. If f ( x) does not change sign at c, then
relative maximum.
f (c) is neither a relative minimum nor a
4. At a left endpoint, if f ( x)  0 you have a relative maximum, and if f ( x)  0 you have
a relative minimum for x  a .
5. At a right endpoint, if f ( x)  0 you have a relative minimum, and if
have a relative maximum for x  a .
f ( x)  0 you
Concavity
Let
f
be differentiable on an open interval I. The graph of
increasing on the interval and concave downward on I if
f
is concave upward on I if
f  is decreasing on the interval.
Test for Concavity
Let
f
1.
If
f ( x)  0 for all x
in I, then the graph of
f
is concave upward in I.
2.
If
f ( x)  0 for all x
in I, then the graph of
f
is concave downward in I.
be a function whose second derivative exists on an open interval I.
Note:
If f ( x)  0 then
f
is linear and concavity is not defined for a line.
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f  is
AP Calculus BC
Student Notes – Chapter 4
Point of Inflection
If
1.
(c, f (c))
is a point of inflection of
f
, then either
f (c)  0
or
2.
f
is not differentiable at
xc
Second Derivative Test
Let f be a function such that
interval containing c, then
f (c)  0 and the second derivative of f
1.
If
f (c)  0 , then f (c) is a relative minimum.
2.
If
f (c)  0 , then f (c) is a relative maximum.
3.
If
f (c)  0 the test fails.
exists on an open
Use the first derivative test.
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