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Section 3.1 Extrema on an Interval
Definition of Extrema
Let f be defined on an interval I containing c.
1. f(c) is the minimum of f on I if f(c) ≤ f(x)
for all x in I.
2. f(c) is the maximum of f on I if f(c) ≥ f(x)
for all x in I.
A min or max of f on I is called the extremum.
Extreme Value Theorem
If f is continuous on [a, b], then f has both a
min and max on [a, b].
Definition of Relative Extrema
1. If there is an open interval containing c on which
f(c) is a maximum, then f(c) is called a relative
maximum of f, or we say "f has a relative max at
2. If there is an open interval containing c on which
f(c) is a minimum, then f(c) is called a relative
minimum of f, or we say "f has a relative min at
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Definition of a Critical Number
Let f be defined at c. If f '(c) = 0 or if f is not
differentiable at c, then c is a critical number
of f.
Theorem 3.2
If f has a relative min or max at x = c, then c
is a critical number of f.
Guidelines to find the extrema of a continuous
function f on [a, b]
1. Find the critical numbers of f in (a, b),
2. Evaluate f at each critical number in (a, b),
3. Evaluate f at each endpoint of [a, b],
4. The least of these values in the min. The
greatest is the max.
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October 24, 2012
Find the critical numbers for
#40 If f(x) = √4 − x2 locate the absolute
extrema over the following intervals.
[-2, 2]
[-2, 0)
(-2, 2)
[1, 2)
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October 24, 2012
Locate the absolute extrema for f over the
given interval.
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Locate the absolute extrema of the function
on the interval [-5, -1].
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#42 Sketch the graph of f and locate the
absolute extrema over the interval [1, 5].
⎧ 2 − x2, 1 ≤ x < 3
f(x) = ⎨
⎩ 2 − 3x, 3 ≤ x ≤ 5
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October 24, 2012
Find the critical numbers of
over the interval 0 ≤ x < 2π.
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