3.2 Extrema and the FirstDerivative Test

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3.2 Extrema and the First-Derivative Test
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Relative Extrema

• Let f be a function defined at c

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Relative Maximum

• f(c) is a relative maximum of f if there exists
  an interval (a,b) containing c such that
  f(x)?f(c) for all x in (a,b)

f(c)
f(x)
c
b
a
x
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Relative Minimum

• f(c) is a relative minimum of f if there exists
  an interval (a,b) containing c such that
  f(x)?f(c) for all x in (a,b)

f(x)
f(c)
c
b
a
x
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Relative Extrema

• f(c) is a relative extremum if f(c) is either a
  relative minimum or a relative maximum
• If f(c) is a relative extremum then c is a
  critical number

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First Derivative Test
f ? lt0
f ? gt0
f ? gt0
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First Derivative Test

• Let c be the only critical number of f in the
  interval (a,b) and f continuous on the interval
  (a,b).
• Determine the sign of f ? to the left of xc and
  to the right of xc. The change in signs
  indicates the following
• From to ? implies f(c) is a relative maximum
• From ? to implies f(c) is a relative minimum
• No change in signs implies f(c) is not a relative
  extremum

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Absolute Extrema

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

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Extreme Value Theorem

• If f is continuous on the closed interval a,b,
  then f takes on both an absolute maximum and
  absolute minimum values on a,b.

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Finding Absolute Extrema

• To find the extrema of a continuous function f on
  the interval a,b
• Evaluate f at each critical number in the
  interval (a,b).
• Evaluate f(a) and f(b).
• The largest number is the maximum and the
  smallest is the minimum value.

Example
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Thats all folks
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Page 189 29

• Find the absolute extrema of h(t)(t?1)2/3 on the
  interval ?7,2
• Note that h(t) is continuous on this interval
• h ?(t)(2/3)(t ?1)?(1/3)
• The critical number is 1
• h(1)0
• h(?7)4, h(2)1
• Abs Max 4, Abs Min 0

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