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Unformatted text preview: c Kendra Kilmer February 28, 2012 Section 4.2 - Maximum and Minimum Values
Deﬁnition The number f (c) is a
• local maximum value of f if when x is near c. • local minimum value of f if when x is near c. Example 1: For what values of x does f (x) have a local extrema? f(x)
a b c d e x Deﬁnition: A critical number of a function f is a number c in the domain of f such that either f ′ (c) = 0 or f ′ (c)
does not exist.
Example 2: Find the critical numbers of each of the given functions.
a) f (x) = x5 + 5x4 − 75x3 + 7 b) g(x) = x2/5 − x−3/5 1 c Kendra Kilmer February 28, 2012 Deﬁnition Let c be a number in the domain D of a function f . Then f (c) is the
• absolute maximum value of f on D if
• absolute minimum value of f on D if for all x in D.
for all x in D. Example 3: Find the absolute maximum and absolute minimum of each function if they exist. a) b) c)
Example 4: Find the absolute max and min for the functions in Example 1 on the interval [−1, 1] The Closed Interval Method To ﬁnd the absolute maximum and minimum values of a continuous function f on a
closed interval [a, b]:
1. Find the values of f at the critical numbers of f in (a, b).
2. Find the values of f at the endpoints of the interval.
3. The largest of the values from Steps 1 and 2 is the absolute maximum values; the smallest of these values is the
absolute minimum value.
2 c Kendra Kilmer February 28, 2012 Example 5: Find the absolute extrema of the following functions on the given intervals
a) f (x) = 2x3 − 3x2 − 12x + 24 on [1, 4] b) g(x) = x3 + 3x2 − 9x − 7 on [−2, 0] c) h(x) = x ln x on [0.1, 5] Section 4.2 Highly Suggested Homework Problems: 1, 3, 7, 9, 13, 21, 25, 29, 41, 45, 47, 49, 51 3 ...
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- Fall '08