4.1 - Maximum and Minimum Values
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1 4.1 - Maximum and Minimum Values Calculus I, Section 011 Zachary Cline Temple University October 27, 2017
2 Maximum and Minimum Values absolute max. of 5 occurs at 3 absolute min. of 2 occurs at 6
3 Maximum and Minimum Values abs. min. at a local max. at b local min. at c abs. and local max. at d local min. at e
4 Example y = cos x
5 What are the absolute/local extreme values?
6 What are the absolute/local extreme values? local and absolute min. of 0 at 0.
7 What are the absolute/local extreme values?
8 What are the absolute/local extreme values? (No minima or maxima)
9 What are the absolute/local extreme values?
10 What are the absolute/local extreme values? abs. max. of 37 at 1
11 What are the absolute/local extreme values? abs. max. of 37 at 1 local min. of 0 at 0
12 What are the absolute/local extreme values? abs. max. of 37 at 1 local min. of 0 at 0 local max. of 5 at 1
13 What are the absolute/local extreme values? abs. max. of 37 at 1 local min. of 0 at 0 local max. of 5 at 1 local and abs. min. of 27 at 3
14 Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b].
15 Extreme Value Theorem If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b]. To see why continuity on the closed interval is necessary, consider the following:
16 Fermat s Theorem If f has a local maximum or minimum at c, then f (c) = 0 or f (c) does not exist.
17 Fermat s Theorem If f has a local maximum or minimum at c, then f (c) = 0 or f (c) does not exist. Example of an extreme value when f (c) does not exist:
18 Fermat s Theorem If f has a local maximum or minimum at c, then f (c) = 0 or f (c) does not exist. Example of an extreme value when f (c) does not exist: Even if f (c) = 0, there might not be a local max. or min. at c:
19 Exercise 1 Find the critical numbers of f(x) = x 3 5 (4 x).
20 The Closed Interval Method Let f be a continuous function on a closed interval [a, b]. To find the absolute minimum and maximum values of f:
21 The Closed Interval Method Let f be a continuous function on a closed interval [a, b]. To find the absolute minimum and maximum values of f: 1. Find the values of f at critical numbers in (a, b).
22 The Closed Interval Method Let f be a continuous function on a closed interval [a, b]. To find the absolute minimum and maximum values of f: 1. Find the values of f at critical numbers in (a, b). 2. Find the values of f at the endpoints, a and b.
23 The Closed Interval Method Let f be a continuous function on a closed interval [a, b]. To find the absolute minimum and maximum values of f: 1. Find the values of f at critical numbers in (a, b). 2. Find the values of f at the endpoints, a and b. 3. The largest of all these is the absolute maximum. The smallest is the absolute minimum.
24 Example Find the absolute maximum and minimum values of the function f(x) = x 3 3x 2 + 1, 1 2 x 4
25 Example Find the absolute maximum and minimum values of the function f(x) = x 3 3x 2 + 1, 1 2 x 4
26 Exercise 2 Find the absolute maximum and minimum values of the function f(x) = x x2 6x + 2, 0 x 2
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