Download Section 3.1 Extrema on an Interval Definition of Relative Extrema

November 14, 2013
Section 3.1 Extrema on an Interval
Definition of Extrema
Let f be defined on an interval I containing c.
1. f(c) is the minimum of f on I if f(c) ≤ f(x)
for all x in I.
2. f(c) is the maximum of f on I if f(c) ≥ f(x)
for all x in I.
A min or max of f on I is called the extremum.
Extreme Value Theorem
If f is continuous on [a, b], then f has both a
min and max on [a, b].
Definition of Relative Extrema
1. If there is an open interval containing c on which
f(c) is a maximum, then f(c) is called a relative
maximum of f, or we say "f has a relative max at
2. If there is an open interval containing c on which
f(c) is a minimum, then f(c) is called a relative
minimum of f, or we say "f has a relative min at
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Definition of a Critical Number
Let f be defined at c. If f '(c) = 0 or if f is not
differentiable at c, then c is a critical number
of f.
Theorem 3.2
If f has a relative min or max at x = c, then c
is a critical number of f.
Guidelines to find the extrema of a continuous
function f on [a, b]
1. Find the critical numbers of f in (a, b),
2. Evaluate f at each critical number in (a, b),
3. Evaluate f at each endpoint of [a, b],
4. The least of these values in the min. The
greatest is the max.
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Section 3.2
Rolle's Theorem and the
Mean Value Theorem
Rolle's Theorem
Let f be continuous on [a, b] and differentiable
on (a, b). If f(a) = f(b), then there is at least one
number c in (a, b) such that f '(c) = 0.
f(b) = f(a)
a
c
b
The Mean Value Theorem
If f is continuous on [a, b] and differentiable
on (a, b), then there exists a number c in (a, b)
such that
f(b) − f(a)
f '(c) =
b−a
f(b)
f(a)
a
b
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Section 3.3 Increasing and Decreasing
Functions and the First Derivative Test
Definitions of Increasing and Decreasing Functions
A function f is increasing on an interval if for any
two numbers a and b in the interval, a < b implies
f(a) < f(b).
A function f is decreasing on an interval if for any
two numbers a and b in the interval, a < b implies
f(a) > f(b).
f is increasing on (a, b)
•
f(b)
f(a)
•
a
f (a)
b
f is decreasing on (a, b)
•
•
f (b)
a
b
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Test for Increasing and Decreasing Functions
Let f be a function that is continuous on [a, b]
and differentiable on (a, b).
1. If f '(x) > 0 for all x in (a, b), f is increasing
on [a, b].
2. If f '(x) < 0 for all x in (a, b), f is decreasing
on [a, b].
3. If f '(x) = 0 for all x in (a, b), f is constant
on [a, b].
The First Derivative Test
Let c be a critical number of a function f that
is continuous on an open interval I containing
c. If f is differentiable on the interval, except
possibly at c, then f(c) can be classified as
follows.
1. If f '(x) changes from negative to positive
at c, then f has a relative minimum at (c, f(c)).
2. If f '(x) changes from positive to negative
at c, then f has a relative maximum at (c, f(c)).
3. If f '(x) is positive on both sides of c or
negative on both sides of c, then f(c) is neither
a relative minimum nor a relative maximum.
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Section 3.4 Concavity and the Second
Derivative Test
Definition of Concavity
Let f be differentiable on an open interval
I. The graph of f is concave upward on I if f ' is
increasing on the interval and concave downward
on I if f ' is decreasing on the interval.
Test for Concavity
Let f be a function whose second derivative
exists on an open interval I.
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.
y = f (x)
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Definition of Point of Inflection
Let f be function that is continuous on an
open interval and let c be a number in this interval.
If the graph of f has a tangent line at this point
(c, f (c)), then this point is a point of inflection of
the graph of f if the concavity of f changes from
upward to downward (or downward to upward)
at the point.
Points of Inflection
If (c, f (c)) is a point of inflection of the graph
of f, then either f "(c) = 0 or f " does not exist at
x = c.
The Second Derivative Test
Let f be a function such that f '(c) = 0 and the
second derivative of f exists on an open interval
containing c.
1. If f "(c) > 0, then f has a relative min at (c, f (c)).
2. If f "(c) < 0, then f has a relative max at (c, f (c)).
If f "(c) = 0, the test fails. Revert to using the First
Derivative Test in this case.
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6. Find values of a, b, c, and d so that
f (x) = ax3 + bx2 + cx + d has an inflection point at
(1, -1) and a relative extremum at (0, 3).
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