Maximum and Minimum Values - 3.3
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1 Maimum and Minimum Values Critical Numbers Definition A point c in the domain of f is called a critical number offiff c or f c is not defined. Eample a. The graph of f is given below. Find all possible critical numbers of f for in,. b. The graph of g is give below. Suppose D g :, 4. Find all possible critical numbers of g a. f b. g a. Find graphicall all in D f at which f :.8,.8,., 4. Find graphicall all in D f at which f is not defined:. Answer: Critical numbers of f :.8,.8,., 4and. b. Find graphicall all in D g at which g :,,, 4. Find graphicall all in D g at which g is not defined: 3. Answer: Critical numbers of f are,,, 4, and 3. Eample Let f 3 3. Find all critical numbers of f. D f, Compute f : f Find all in D f at which f : f when, and. Find all in D f at which f is not defined: None. Answer: Critical numbers of f are and. Eample Let f e. Find all critical numbers. 4
2 D f ;, Compute f : f e 4 e 4 e 4 4 Find all in D f at which f : f when Find all in D f at which f is not defined: f is not defined when 4,. However, are not in D f so the are not critical points of f. 7 Answer: Critical numbers of f are. Eample Let f D f,. Compute f :. Find all critical numbers. f 4 8 Find all in D f at which f : f when 8 3/ 3/ 8, 8 /3 4 Find all in D f at which f is not defined: f is not defined when and isind f. Answer: Critical numbers of f are and. 4. Local and Absolute Etrema: Definition Let c be a point in the domain of f. a. f c is a local maimum of f if f c f for all near c. The number c is called a local maimum point. b. f c is a local minimum of f if f c f for all near c. The number c is called a local minimum point. In either case, f c is called a local etremum of f (its plural form is local etrema). Definition Let c be a point in the domain of f. a. f c is an absolute maimum of f if f c f for all in D f. The number c is called the absolute maimum point. b. f c is an absolute minimum of f if f c f for all in D f. The number c is called the absolute minimum point. In either case, f c is called an absolute etremum of f (its plural form is absolute etrema). Graphicall, it is eas to classif a local etremum and an absolute etremum. Eample Consider the graph in Figure 3. on Page 6.
3 Local maimum points: b, d Local minimum points: a, c Absolute maimum point: d Absolute minimum point: c Observations: Let f be continuous on a, b and c in a, b be a critical number of f. a. If f changes sign at c from positive to negative (f changes from increasing to decreasing) then f c is a local maimum. b. If f changes sign at c from negative to positive (f changes from decreasing to increasing) then f c is a local minimum. Theorem (Fermat s Theorem) Suppose that f c is a local etremum (either local minimum or local maimum). Then c must be a critical number of f. Note that the fact that c is a critical number of f DOES NOT impl f c is a local etremum. Steps for finding local maimum and local minimum values: a. Compute f. b. Find all critical numbers of f (zeros of f ), sa, c,..., c k. c. Check the signs of f before and after each c i : i. if the sign of f changes from to at c i, then f c i is a local maimum value of f ; and ii. if the sign of f changes from to at c i, then f c i is a local minimum value of f. Theorem (Etreme Value Theorem) Let f be continuouson a, b. Then f attains an absolute maimum and an absolute minimum on a, b. The absolute etrema of f must occur at an endpoint (a or b) or at a critical number c. Steps for finding absolute maimum and absolute minimum values: a. Compute f. b. Find all critical numbers of f (zeros of f ), sa, c,..., c k. c. Compare values of f c, f c,..., f c k, f a and f b, and i. the maimum value of these numbers is the maimum value of f;and ii. the minimum value of these numbers is the minimum value of f. Eample Find critical numbers of f and determine if f has a local maimum or local minimum or neither at each of the critical numbers. a. f 3 b. f c. f sincos a. f 3 3
4 f 3,. f has onl one critical number at. Since f 3 does not change sign, so f is neither a local maimum nor a local minimum. b. f f 4 4 4, 4 Check the sign change of f at and 4. f 3 f changes from to at 4, f 3 f 4 is a local maimum., f changes from to at, f 4 f is a local minimum. 3 Check with the graphs of f in a. andinb.: c. f sincos,,, D f,, D f, i. f cos sin ii. f, cos sin, cos sin, tan, tan tan, 4, 4 4, tan, 3 4, , 3 4, 4, 7 4 iii. Check sign change of f : are critical numbers. f, f f, f 3, f 4
5 interval,, 3 3,, 7 7, sign of f f and f are local maimum values; and f 3 and f 7 are local minimum values. Eample a Giveafunctionf that has no absolute maimum but absolute minimum over,. b Giveafunctionf that has no absolute minimum but absolute maimum over,. c Giveafunctionf that has no absolute maimum and minimum over,. a f 3 b g 3 c h ln f 3 g 3 h ln Eample Give a function f which has no absolute maimum on, f Eample Give a function f which has no etrema on,.
6 f Eample Let f 3 3. Find the absolute etrema of f on the interval, 4. Sketch the graph of f for 4: f From the graph of f, we see the absolute maimum occurs at 4 and the absolute minimum occurs at one of two critical numbers of f in, 4. Now let us locate the absolute maimum and absolute minimum algebraicall. a. Find all critical numbers of f : f ,, b. Check values f, f 4, f and f : f, f 4 37, f, f Conclusion: On the interval, 4, the absolute maimum of f is 37when 4 and the absolute minimum is when. 6
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