Maxima and minima of functions

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Maxima and minima of functions

• Lesson 2.2

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Definitions

• Global extrema If f(c) f(x) for all x in the
  domain of f, f(c) is the global maximum value of
  f. If f(c) f(x) for all x in the domain of f,
  f(c) is the global minimum value of f.
• Local extrema If f(c) f(x) for all x in some
  open interval containing c, f(c) is a local
  maximum value of f. If f(c) f(x) for all x in
  some open interval containing c, f(c) is a local
  minimum value of f.

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Example 1
Y button

• Approximate the global and local maximum and
  minimum on each given domain for the function k
  defined by
• k(x) -2x4 3x3 4x2 5x 5
• Set of all real numbers
• -1 x 1
• x lt -2

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• k(x) -2x4 3x3 4x2 5x 5
• Set of all real numbers
• -no global min, local min. at x .477, k(x)
  3.747
• - local/global max. at x -.865, k(x) 9.257,
• - local max. at x 1.513, k(x) 6.502

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• k(x) -2x4 3x3 4x2 5x 5
• -1 x 1
• -local/global min at x .477, k(x) 3.747
• -local min at endpoint x-1, k(x) 9
• -local/global max. at x -.865, k(x) 9.257,
• - local max at endpoint, x 1, k(x) 5

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• k(x) -2x4 3x3 4x2 5x 5
• x lt -2
• - there is no minimum since the function
  decreases without bound on the interval (-8, -2).
• -There is no maximum because k(x) increases as
  x increases and there is no greatest value of x
  on this interval.

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Find the extrema of f(t)2t4 4t 1
Over 0,8) - local/global min t0, f(t) 1 -
No local/global max. Over (-3,1) -local/global
min t-.787, f(t) -1.381 -No local/global
max.
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Homework

• Page 91
• 3, 5, 6, 7
• 10, 12, 14