( ) = 0. ( ) does not exist. 4.1 Maximum and Minimum Values Assigned videos: , , , DEFINITION Critical number

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1 4.1 Maximum and Minimum Values Assigned videos: , , , DEFINITION Critical number A critical number of a function f is a number c in the domain of f such that f c or f c ( ) does not exist. ( ) = 0

2 EXAMPLE Find the critical numbers for each function. f ( x) = x 3 + x 2 x f ( x) = xe x2 8 ( ) = x 1 f x ( )2 x +1

3 DEFINITIONS Absolute maximum, absolute minimum Suppose f(a) f(x) for all x in the domain of f. Then f has an absolute maximum at a, and the number f(a) is the maximum value of f. Likewise, f has an absolute minimum at a if f(a) f(x) for all x in the domain of f, and the number f(a) is the minimum value of f. Informally, if f has an absolute maximum at a, then there is no point on the graph of f(x) that is higher than the point (a, f(a)). If f has an absolute minimum at a, then there is no point on the graph of f(x) that is lower than the point (a, f(a)).

4 DEFINITION Local maximum, local minimum Suppose f(a) f(x) for all x in some open interval containing a. Then f has an local (or relative) maximum at a, and the number f(a) is a local maximum value of f. Local or relative minimum is defined similarly. Informally, if f has a local maximum at a, then the point (a, f(a)) is higher than all other nearby points both to the left and the right on the curve, but there may be points elsewhere on the curve that are higher; similarly for a local minimum.

5 EXAMPLE Identify all local and absolute extrema of the function f whose graph is shown below. Note that the domain of this continuous function is [-8, 18]

6 FACT If f has a local maximum or minimum at c, then c is a critical number of f. FACT (The Extreme Value Theorem) A continuous function f will have both an absolute maximum and an absolute minimum on the closed interval [a, b]. These absolute extrema could occur at the endpoints of the interval, or at critical numbers within the interval.

7 METHOD (Based on the Extreme Value Theorem) To find the absolute maximum and absolute minimum of a continuous function f on the closed interval [a, b]: 1. Find the values of f at all 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 will be the absolute maximum. The smallest of those values will be the absolute minimum.

8 EXAMPLE Find the absolute minimum and absolute maximum of f ( x) = x 3 + x 2 x on the interval [ 2, 0] Note: earlier today, we found that this function has two critical numbers: 1/3 and 1. Of these two critical numbers, one of them ( 1) is in the interval ( 2, 0), so it is the only critical number that matters for this exercise. We evaluate f( 2) f(0) f( 1) The largest of these three values will be the absolute max, the smallest will be the absolute min.

9 EXAMPLE Find the absolute minimum and absolute maximum of ( )2 ( ) = x 1 f x x +1 on the interval [ 6, 2]. Note: earlier, we found that this function has two critical numbers: 1 and 3. Of these critical numbers, only 3 is in the interval ( 6, 2), so it is the only critical number that matters for this exercise. Our function is continuous for all real numbers except x = 1, so, in particular, f is continuous on our closed interval [ 6, 2]. According to the Extreme Value Theorem, f will have an absolute maximum and an absolute minimum on this interval. We find them by evaluating f( 6), f( 2), and f( 3).

10 EXAMPLE Let f ( x) = x x 1 3. What is the domain? Find all critical numbers.