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Unformatted text preview: Math 006 (Lecture 28) Absolute Maxima and Minima
Deﬁnition 1. Absolute Maxima and Minima 1. If f (c) ≥ f (x) for all x in the domain of f , then f (c) is called the absolute maximum value of f . 2. If f (c) ≤ f (x) for all x in the domain of f , then f (c) is called the absolute minimum value of f . Example 1. Find the absolute maxima and minima of the following functions x3 (a) f (x) = − 4x (b) g (x) = 4 − x2 (c) h(x) = x2/3 . 3 Theorem 1. A function that is continuous on the interval [a, b] (a ≤ x ≤ b) has both an absolute maximum value and an absolute minimum value on that interval. The absolute extrema must always occur at 1. critical values, or 2. the endpoints Example 2. Find the absolute maximum and absolute minimum values of f (x) = x3 − 12x on each of the following intervals: (A) [−5, 5] (B) [−3, 3] (C) [−3, 1] 1 Second Derivative and Extrema
Example 3. Consider (A) f (x) = x2 (B) g (x) = −x2 (C) h(x) = x3 Theorem 2. Let c be the only critical value of f (x). 1. If f (c) > 0, then f (c) is an absolute minimum. 2. If f (c) < 0, then f (c) is an absolute maximum. 3. If f (c) = 0, we have no conclusion. Example 4. Find the absolute maximum of each function on 0 < x < ∞ (a) f (x) = x + 4 x (b) f (x) = (ln x)2 − 3 ln x 2 ...
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- Fall '09