Calculus 5.1: Extrema on an Interval

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Calculus 5.1Extrema on an Interval

• Using derivatives to analyze extrema of a
  function.

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Definition of Extrema

• The minimum and maximum of a function on an
  interval are called the extreme values, or
  extrema, of the function on the interval.
• Extrema can occur at interior points or endpoints
  of an interval.

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The Extreme Value Theorem

• If f is continuous on a closed interval a,b,
  then f has both a minimum and a maximum on the
  interval.

(This theorem is an example of an existence
theorem because it tells of the existence of
minimum and maximum values, but does not show how
to find these values.)
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Relative Extrema

• The relative maximum and relative minimum on a
  graph are the hills and valleys of the graph.
• If the hill (or valley) is smooth and rounded,
  the graph has a horizontal tangent at this point
  and the derivative is zero here.
• If the hill (or valley) is sharp and peaked, the
  function is not differentiable at this point and
  the derivative is undefined here.

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Example 1
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Definition of Critical Number

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

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Theorem 5.2

• The relative extrema of a function occur only at
  the critical numbers of the function.
• Knowing this, you can use the following
  guidelines to find extrema on a closed interval.

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Guidelines for Finding Extrema on a Closed
Interval

• To find the extrema of a continuous function on
  a closed interval a,b, use the following steps.
• Find the critical numbers of f in (a,b).
• Evaluate f at each critical number in (a,b).
• Evaluate f at each end point of a,b.
• The least of these values is the minimum. The
  greatest is the maximum.

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Example 2

• Locate the absolute extrema of the function
• on the interval -2,3.

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Example 3
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assignment

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