Math 101

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Math 101

• Mathematics for Management Students

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Differentiation5Increasing and Decreasing
Functions,Extreme Values

• Lecture 10

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Increasing and Decreasing Functions
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Example

• The following functions are increasing on the
  given intervals,

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Example

• The following functions are decreasing on the
  given intervals,

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Test for Increasing and Decreasing Functions

• Let f be differentiable on the interval (a, b),
  then
• 1. If f ' (x) gt 0 for all x in (a, b), then f is
  increasing on (a, b)

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Test for Increasing and Decreasing Functions

• Let f be differentiable on the interval (a, b),
  then
• If f ' (x) gt 0 for all x in (a, b), then f is
  increasing on (a, b)
• 2. If f ' (x) lt 0 for all x in (a, b), then f is
  decreasing on (a, b)

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Test for Increasing and Decreasing Functions

• Let f be differentiable on the interval (a, b),
  then
• 1. If f ' (x) gt 0 for all x in (a, b), then f is
  increasing on (a, b)
• 2. If f ' (x) lt 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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Example
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Solution
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Critical Numbers and Their Use

• If f is defined at c,
• then c is a critical number of f if
• f ' (c) 0 or
• 2) f ' is undefined at c.

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Critical Numbers and Their Use

• The critical numbers are used to find the open
  intervals on which the function is increasing or
  decreasing using the following procedure

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Guidelines for applying Increasing/decreasing Test

• 1.Find the derivative of f

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Guidelines for applying Increasing/decreasing Test

• 2. Locate the critical numbers of f and use
• these numbers to determine test intervals.
• that is, find all x for which f ' (x) 0 or
  f(x) is undefined.

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Guidelines for applying Increasing/decreasing Test

• 3. Test the sign of f ' (x) at an arbitrary
  number in each test intervals.

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Guidelines for applying Increasing/decreasing Test

• 4. Use the test for increasing and decreasing
  functions to decide whether f is increasing or
  decreasing on each interval.

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Example

• Find the open intervals, on which the function is
  increasing or decreasing,

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Solution

• First of all we note that the function is a
  polynomial, so the domain is R
• We find f ' (x),
• f ' (x) 3x2 - 3x,

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Solution

• To get the critical numbers,
• we set f ' (x) 0 and solve for x

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Solution

• There are no values for which f ' is undefined.
• Then x 0 and x 1are the only critical
  numbers.
• So, the intervals we need to test are (-
  , 0), (0, 1), (1, ).

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The table summarizes the test of these three
intervals
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Example

• Find the open intervals, on which the function is
  increasing or decreasing,

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Classification of Extreme Points Using the First
derivative Test

• Let f be continuous on the interval (a, b) in
  which c is the only critical number.
• If f is differentiable on the interval (except
  possibly at c), then f(c) can be classified as
• a relative minimum,
• a relative maximum,
• neither,
• as shown.

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Relative Minimum

• On the interval (a, b), if f ' (x) is negative
  to the left of x c and positive to the right of
  x c, then f(c) is a relative minimum.

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Relative maximum.

• On the interval (a, b), if f ' (x) is positive to
  the left of x c and negative to the right of x
  c, then f(c) is a relative maximum.

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Neither

• On the interval (a, b), if f ' (x) has the same
  sign to the left and right of x c, then f(c) is
  not a relative extremum of f

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Example

• Find the relative extrema of the function
• f(x) 2x3 - 3x2 - 36x 14

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Solution

• Begin by finding the critical numbers of f

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Solution

• Because f ' (x) is defined for any real number,
  then the only critical numbers of f are -2, 3
• Using these numbers, you can form the test for
  the three intervals (- ,-2), (-2, 3), (3, ).

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The tests results of these intervals are shown in
the table
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Solution

• The results show that
• -2 is a relative maximum.
• 3 is a relative minimum.

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Absolute Extrema

• Let f be defined on an interval I containing c.
• f(c) is an absolute minimum of f on I if
  f(c) f(x) for every x in I.
• 2. f(c) is an absolute maximum of f on I if
  f(c) f(x) for every x in I

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Note

• The absolute minimum and absolute maximum values
  of a function on an interval are sometimes simply
  called the minimum and maximum.