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Fig. 143 it is clear that f has neither a relative maximum nor a relative minimum at x = 0. in .NET framework
Fig. 143 it is clear that f has neither a relative maximum nor a relative minimum at x = 0. Scanning QR Code JIS X 0510 In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. QR Code Printer In .NET Using Barcode generator for .NET Control to generate, create QRCode image in .NET applications. EXAMPLE Consider the function f (x) = x3 . Because f (x) = 3x2 , f (x) if and only if x = 0. But from the graph of f in QR Code ISO/IEC18004 Reader In VS .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications. Drawing Bar Code In Visual Studio .NET Using Barcode generator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. In 23, a method will be given that often will enable us to determine whether a relative extremum actually exists when f (c) = 0. Decode Bar Code In Visual Studio .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Paint QR Code In C#.NET Using Barcode generation for VS .NET Control to generate, create QR image in .NET framework applications. 14.2 ABSOLUTE EXTREMA Practical applications usually call for nding the absolute maximum or absolute minimum of a function on a given set. Let f be a function de ned on a set E (and possibly at other points, too), and let c belong to E . Then f is said to achieve an absolute maximum on E at c if f (x) f (c) for all x in E . Similarly, f is said to achieve an absolute minimum on E at d if f (x) f (d) for all x in E . If the set E is a closed interval [a, b], and if the function f is continuous over [a, b] (see Section 10.3), then we have a very important existence theorem (which cannot be proved in an elementary way). Theorem 14.2: (ExtremeValue Theorem): Any continuous function f over a closed interval [a, b] has an absolute maximum and an absolute minimum on [a, b]. Encoding QR In .NET Using Barcode generation for ASP.NET Control to generate, create QR Code image in ASP.NET applications. QRCode Creator In VB.NET Using Barcode printer for .NET Control to generate, create QR Code ISO/IEC18004 image in .NET framework applications. MAXIMUM AND MINIMUM PROBLEMS
Creating Code 128B In Visual Studio .NET Using Barcode drawer for .NET Control to generate, create Code 128C image in VS .NET applications. UPCA Supplement 5 Creator In .NET Framework Using Barcode printer for .NET Control to generate, create UCC  12 image in VS .NET applications. [CHAP. 14
Barcode Generation In VS .NET Using Barcode encoder for VS .NET Control to generate, create bar code image in .NET framework applications. Leitcode Drawer In .NET Using Barcode creation for VS .NET Control to generate, create Leitcode image in Visual Studio .NET applications. EXAMPLES
EAN13 Encoder In Java Using Barcode generation for BIRT reports Control to generate, create EAN13 image in BIRT applications. UPCA Supplement 2 Reader In Visual Studio .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. (a) Let f (x) = x + 1 for all x in the closed interval [0, 2]. The graph of f is shown in Fig. 144(a). Then f achieves an absolute maximum on [0, 2] at x = 2; this absolute maximum value is 3. In addition, f achieves an absolute minimum at x = 0; this absolute minimum value is 1. (b) Let f (x) = 1/x for all x in the open interval (0, 1). The graph of f is shown in Fig. 144(b). f has neither an absolute maximum nor an absolute minimum on (0, 1). If we extended f to the halfopen interval (0, 1], then there is an absolute minimum at x = 1, but still no absolute maximum. x + 1 if 1 x < 0 (c) Let f (x) = 0 if x = 0 x 1 if 0 < x 1 See Fig. 144(c) for the graph of f . f has neither an absolute maximum nor an absolute minimum on the closed interval [ 1, 1]. Theorem 14.2 does not apply, because f is discontinuous at 0. Scanning Code39 In Java Using Barcode scanner for Java Control to read, scan read, scan image in Java applications. Printing Code 3 Of 9 In None Using Barcode creator for Office Excel Control to generate, create Code 3 of 9 image in Office Excel applications. Fig. 144 Printing USS Code 39 In ObjectiveC Using Barcode creator for iPhone Control to generate, create Code 39 Extended image in iPhone applications. Bar Code Recognizer In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Critical Numbers To actually locate the absolute extrema guaranteed by Theorem 14.2, it is useful to have the following notion. De nition: A critical number of a function f is a number c in the domain of f for which either f (c) = 0 or f (c) is not de ned. EXAMPLES Barcode Creation In None Using Barcode drawer for Excel Control to generate, create bar code image in Excel applications. Drawing Bar Code In Java Using Barcode maker for Java Control to generate, create bar code image in Java applications. (a) Let f (x) = 3x 2 2x + 4. Then f (x) = 6x 2. Since 6x 2 is de ned for all x, the only critical numbers are given by 6x 2 = 0 6x = 2 2 1 x= = 6 3 Thus, the only critical number is 1 . 3 CHAP. 14] MAXIMUM AND MINIMUM PROBLEMS
(b) Let f (x) = x 3 x 2 5x + 3. Then f (x) = 3x 2 2x 5, and since 3x 2 2x 5 is de ned for all x, the only critical numbers are the solutions of 3x 2 2x 5 = 0 (3x 5)(x + 1) = 0 3x 5 = 0 or x + 1 = 0 x = 1 x = 1 3x = 5 or 5 or x= 3 Hence, there are two critical numbers, 1 and 5 . 3 (c) Let f (x) = x. Thus, f (x) = x if 0 x x if x < 0. We already know from the example in Section 13.1 that f (0) is not de ned. Hence, 0 is a critical number. Since Dx (x) = 1 and Dx ( x) = 1, there are no other critical numbers. Method for Finding Absolute Extrema Let f be a continuous function on a closed interval [a, b]. Assume that there are only a nite number of critical numbers c1 , c2 , . . . , ck of f inside [a, b]; that is, in (a, b). (This assumption holds for most functions encountered in calculus.) Tabulate the values of f at these critical numbers and at the endpoints a and b, as in Table 141. Then the largest tabulated value is the absolute maximum of f on [a, b], and the smallest tabulated value is the absolute minimum of f on [a, b]. (This result is proved in Problem 14.1.) Table 141 x c1 c2 . . . ck a b f (x) f (c1 ) f (c2 ) . . . f (ck ) f (a) f (b)

