B.2.3 - Extreme Values

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B.2.3 - Extreme Values

• Calculus - Mr Santowski

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Lesson Objectives

• 1. Use Calculus methods to determine the absolute
  and relative extrema of a continuous
  differentiable function
• 2. State the extreme value theorem
• 3. Apply concepts of increase, decrease and
  critical numbers and absolute extrema to a real
  world problem

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Fast Five

• 1. Determine the x coordinates of the critical
  point(s) of f(x) x4 - 18x2 1
• 2. On the restricted domain of 0,4, find the
  function values at the end points if
• 3. Evaluate
• 4. If , determine whether or
  not the function has an extreme point.

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(A) Terms

• Given a function, f(x), that is defined on a
  given interval and let c be a number in the
  domain
• f(c) is the ABSOLUTE or GLOBAL maximum of f(x) on
  the interval if f(c) gt f(x) for every x in the
  interval
• Now, sketch an example of what has just been
  described.

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(A) Terms

• Given a function, f(x), that is defined on a
  given interval and let c be a number in the
  domain
• f(c) is the ABSOLUTE or GLOBAL minimum of f(x) on
  the interval if f(c) lt f(x) for every x in the
  interval
• Now, sketch an example of what has just been
  described.

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(A) Terms

• A FUNCTION is said to have an absolute extremum
  (or extrema) at x c if it either has either an
  absolute maximum or an absolute minimum at x c

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(B) Extreme Value Theorem

• A function, f(x), that is CONTINUOUS on a CLOSED
  interval a,b will have BOTH an absolute maximum
  value and an absolute minimum value on the closed
  interval
• Now, sketch an example of what has just been
  described.

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(C) Extrema and Open Intervals

• We now consider the importance of an open vs
  closed interval using the functions f(x) x3 and
  g(x) ln(x)

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(C) Extrema and Open Intervals

• To illustrate the point about intervals, consider
  the functions, f(x) x3 and g(x) ln(x)
• If we have an open interval (-?,?) for f(x) and
  (0,?) for g(x), then we should consider the end
  behaviours of the two functions
• For f(x) x3, limx??, f(x)?? and limx ?-?,
  f(x)?-?
• For g(x) ln(x), limx?0, g(x)?-? and limx??,
  g(x)??
• So, on OPEN intervals, absolute extrema MAY NOT
  exist

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(D) Terms Relative or Local

• We say that f(x) has a relative (or local)
  maximum at x c if f(x) lt f(c) for every x in
  some open interval around x c.
• Now, sketch an example of what has just been
  described.
• We say that f(x) has a relative (or local)
  minimum at x c if f(x) gt f(c) for every x in
  some open interval around x c.
• Now, sketch an example of what has just been
  described.

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(E) Terms - Diagram

• To understand the terms, a visualization will
  help

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(F) Example

• Problem solving approach for finding absolute
  extrema of f(x) on a,b.
• 1. Verify that the function is continuous on
  a,b.
• 2. Find all critical points of f(x) that are in
  the interval a,b. This makes sense if you
  think about it. Since we are only interested in
  what the function is doing in this interval we
  dont care about critical points that fall
  outside the interval.
• 3. Evaluate the function at the critical points
  found in step 1 and the end points.
• 4. Identify the absolute extrema.

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(F) Example 1

• Determine the absolute extrema for the following
  function and interval
• g(x) 2x3 3x2 12x 4 on -4,2

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(F) Example 2

• Suppose that the population (in thousands) of a
  certain kind of insect after t months is given by
  P(t) 3t sin(4t) 100
• Determine the minimum and maximum population in
  the first 4 months.

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(F) Example 3

• Suppose that the amount of money in a bank
  account after t years is given by,
• Determine the minimum and maximum amount of money
  in the account during the first 10 years that it
  is open.

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(G) Example

• Find the absolute extrema of g(x) sin(x)cos(2x)
  on the interval -3?/4, ?/3
• Step 1 gt determine d/dx g(x) in order to work
  toward the critical numbers
• Step 2 gt determine max/min values
• Step 3 gt evaluate function values at end points
• Step 4 gt the largest value found in steps 23 is
  the absolute max and the smallest value is the
  absolute min

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(G) Example

• Find the absolute extrema of g(x) sin(x)cos(2x)
  on the interval -3?/4, ?/3
• To find d/dx g(x)

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(G) Example

• Find the absolute extrema of g(x) sin(x)cos(2x)
  on the interval -3?/4, ?/3 gt find critical
  numbers

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(G) Example

• Find the absolute extrema of g(x) sin(x)cos(2x)
  on the interval -3?/4, ?/3 gt evaluate

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(G) Example

• So our 5 function values are
• And the absolute max is and the
  absolute min is at

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(H) Practice

• Find the absolute extrema of the following
  functions

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(I) Internet Links

• Max Min Values from Paul Hawkins at Lamar U
• Extreme Values from Visual Calculus
• Extreme Value Theorem from Pink Monkey

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(J) Homework

• Textbook, S6.1, p331
• (1) Graphs, Q1-8
• (2) Algebra, Q10-32 as needed variety

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(D) Extrema and Continuity

• We now consider the importance of continuity
  using the functions y tan(x) and y 1/x2

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(D) Extrema and Continuity

• If we have discontinuity in y tan(x) at x p/2
  and at x 0 for g(x) 1/x2, then we should
  consider the behaviours of the two functions at
  the discontinuity
• For f(x) tan(x), limx?p/2-, f(x)?-? and
  limx?p/2, f(x)??
• For g(x) 1/x2, limx?0, g(x)?? and limx?0-,
  g(x)??
• Since the function values increase (or decrease)
  without bound (?), there clearly is NO max or
  min value for the function