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Chapter 4 | Applications of Derivatives
and an absolute minimum at x =−1. Hence, f has a local maximum at x =1 and a local minimum at x =−1. (Note that if f has an absolute extremum over an interval I at apoint c that is not an endpoint of I , then f has a local extremum at c .)
Figure 4.18 This function has an absolute maximum and an absolute minimum.
4.12
Find all critical points for f ( x ) = x 3 − 1 2
x 2 −2 x +1.
Locating Absolute Extrema The extreme value theorem states that a continuous function over a closed, bounded interval has an absolute maximum and an absolute minimum. As shown in Figure 4.13 , one or both of these absolute extrema could occur at an endpoint. If an absolute extremum does not occur at an endpoint, however, it must occur at an interior point, in which case the absolute extremum is a local extremum. Therefore, by Fermat’s Theorem , the point c at which the local extremum occurs must be a critical point. We summarize this result in the following theorem. Theorem 4.3: Location of Absolute Extrema Let f be a continuous function over a closed, bounded interval I . The absolute maximum of f over I and the absolute minimum of f over I must occur at endpoints of I or at critical points of f in I .
With this idea in mind, let’s examine a procedure for locating absolute extrema.
Problem-Solving Strategy: Locating Absolute Extrema over a Closed Interval Consider a continuous function f defined over the closed interval [ a , b ]. 1. Evaluate f at the endpoints x = a and x = b . 2. Find all critical points of f that lie over the interval ( a , b ) and evaluate f at those critical points. 3. Compare all values found in (1) and (2). From Location of Absolute Extrema , the absolute extrema must occur at endpoints or critical points. Therefore, the largest of these values is the absolute maximum of f . The smallest of these values is the absolute minimum of f .
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