Extrema On An Interval

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Extrema On An Interval

• Section 3.1

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After this lesson, you should be able to

• Understand the definition of extrema of a
  function on an interval
• Understand the definition of relative extrema of
  a function on an open interval
• Find extrema on a closed interval

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Extrema
Minimum and maximum values on an interval are
called extremes, or extrema on an interval.

• The minimum value of the function on an interval
  is considered the absolute minimum on the
  interval.

• The maximum value of the function on an interval
  is considered the absolute maximum on the
  interval.

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The Extreme Value Theorem
Theorem 3.1 If f is continuous on a closed
interval a, b, then f has both a minimum and a
maximum on the interval.
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Examples

• Given , name any
  extrema of f on the interval
• 0, 5
• (0, 5)

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Relative Extrema

• Relative extrema are turning points of the graph.
• The turning points may occur as smooth hills or
  valleys.

• The turning points may occur as sharp turns.

In these cases, the function is not
differentiable at the relative max/min.
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Critical Numbers

• c is a critical number for f iff
• f(c) is defined (c is in the domain of f)
• f (c) 0 or f (c) does not exist
• If f has a relative max. or relative min, at
• x c, then c must be a critical number for f.
• The (absolute)max and (absolute)min of f on a,
  b occur either at an endpoint of a, b or at a
  critical number in (a, b).

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Finding Extrema on a Closed Interval
To find the max and min of f on a, b

• Find all critical s of f which are in (a, b).
• Find all values of x for which f (c) 0 or f
  (c) does not exist
• Evaluate f at each of the critical values
• Plug each of the critical values into the
  function to find the y-coordinate.
• Evaluate f at each endpoint
• Find f(a) and f(b)
• The smallest value from parts 2 3 is the
  minimum and the largest value from parts 2 3 is
  the maximum of f on a, b.

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Example
Example Find all critical numbers
Domain
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Example
Example Find all critical numbers.
Domain
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Example
Example Find all critical numbers.
Domain
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Example
Example Find the max and min of f on the
interval 0, 4.
Domain
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Example
Example Find the max and min of f on the
interval -1, 1.
Domain
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Example
Example Graph a function f on the interval -3,
4 that has the given characteristics.

• Relative max at x -2
• Absolute min at x 2
• Absolute max at x 4

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Homework
Section 3.1 page 169 1-25 odd, 37, 45