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Clicker Question 1
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What are the critical numbers of
f (x ) = |x + 5| ?
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A. 0
B. 5
C. -5
D. 1/5
E. It has no critical numbers
Clicker Question 2
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What are the critical numbers of
g (x ) = x 2 e4x ?
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A. 0 and -2
B. -1/2
C. 0 and 1/2
D. 0 and -1/2
E. It has no critical numbers.
Finding and Classifying
Extrema (4/1/09 – no fooling)
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In many applications, we are concerned
with finding points in the domain of a
function at which the function takes on
extreme values, either local or global.
To find these, we look at:
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Critical points
Endpoints the domain, if they exist
Behavior as the input variable goes to ∞
Classifying critical points:
The Value Test
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If the values of a function just to the
left and right of a critical point are
below the value at the critical point,
then that point represents a local
maximum of the function.
Similarly for local minimum.
Classifying critical points:
The First Derivative Test
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If the values of the first derivative are
positive to the left but negative to the
right of a critical point, the value of the
function at that point is a local
maximum. (Why??)
Similarly for local minimum.
Classifying critical points:
The Second Derivative Test
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If the value of the second derivative is
negative at a critical point, then the
value of the function at that point is a
local maximum. (Why??)
Similarly for local minimum.
You should use whichever of these tests
is simplest for the given problem.
Clicker Question 3
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Suppose f is a function which has critical
numbers at x = 0, 3, and 6, and suppose
f '(2) = -1 and f '(4) = 1.5. Then at
x = 3, f definitely has a
 A. local maximum
 B. local minimum
 C. global maximum
 D. global minimum
 E. neither a max nor a min
Clicker Question 4
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Suppose f is a function which has critical
numbers at x = 0, 3, and 6, and suppose
f ''(3) = -2. Then at x = 3, f definitely has a
 A. local maximum
 B. local minimum
 C. global maximum
 D. global minimum
 E. neither a max nor a min
Inflection Points
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A point on a function at which the
concavity changes is called an
inflection point of the function
These usually occur at points where the
second derivative is 0 or does not
exists, but not necessarily
(e.g., f (x) = x 4 at 0).
Assignment for Friday
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Read Section 4.3.
In Section 4.3, do Exercises 1, 3, 5, 11,
15, 17, 25, 33, 44, 45, 66.
On Friday, HW#3 will be given out and
will be due Tuesday April 7 at 4:45 pm.