Calculus
Extrema on an Interval
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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 of a function.
Extrema:
 A function has an absolute maximum on an interval at a point x0 if f  x0  is the largest value in the
interval given for the function.
 A function has an absolute minimum on an interval at a point x0 if f  x0  is the smallest value in the
interval given for the function. (Several graphs follow from the text.)
Extreme Value Theorem:
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If a function is continuous on a finite closed interval [a, b] , then the function has both an absolute
maximum and an absolute minimum.
Theorem: If a function has an absolute extremum on an open interval  a, b  , then it must occur at a critical
point of the function.
Calculus
Extrema on an Interval (continued)
Calculus
Extrema on an Interval (continued)
Critical Number: Let a function be defined at a point c. If f   c   0 or if f   c  is not differentiable at c, then c
is a critical number of f. Relative extrema only occur at critical points.
Absolute extrema occurs at endpoints.
Absolute maximum occurs
in the open interval  a, b  at
Absolute maximum occurs in the
open interval  a, b  at a point x0
a point x0 where f   x0   0.
where f is not differentiable.
Calculus
Extrema on an Interval (continued)
Find the value of the derivative, if it exists, at the indicated point.
Calculus
Extrema on an Interval (continued)
Approximate the critical numbers of the function shown in each graph. Determine whether the function has a
relative maximum, relative minimum, absolute maximum, absolute minimum, or none of these at each critical
number on the interval shown.
Find any critical numbers of the function.
Calculus
Extrema on an Interval (continued)
Locate the absolute extrema of the function on the closed interval.
1.
3
x ,  1,1
Class work: Page 165, 11-29 odd
Assignment: Page 165, 33-41 odd, 49-54, page 149,13, 15, 19, 21, 23