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Supplementary Problems in .NET framework
Supplementary Problems Read QR Code JIS X 0510 In .NET Framework Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. QRCode Generator In .NET Using Barcode generation for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. 17.10 Determine whether the hypotheses of Rolle s theorem hold for each function f , and if they do, verify the conclusion of the theorem. (a) f (x) = x 2 2x 3 on [ 1, 3] (c) f (x) = 9x 3 4x on 2 , 2 3 3 (e) f (x) = x2 x 6 on [ 2, 3] x 1 (b) f (x) = x 3 x on [0, 1] (d) f (x) = x 3 3x 2 + x + 1 on [1, 1 + 2] x 3 2x 2 5x + 6 if x = 1 x 1 on [ 2, 3] (f ) f (x) = 6 if x = 1 x2 if 0 x 1 on [0, 2] (h) f (x) = 2 x if 1 < x 2 QR Decoder In .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET applications. Barcode Maker In .NET Framework Using Barcode generation for VS .NET Control to generate, create bar code image in VS .NET applications. (g) f (x) = x 2/3 2x 1/3 on [0, 8] Barcode Reader In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in Visual Studio .NET applications. QR Code Generator In C#.NET Using Barcode creator for .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. 17.11 Verify that the hypotheses of the meanvalue theorem hold for each function f on the given interval, and nd a value c satisfying the conclusion of the theorem. (a) f (x) = 2x + 3 on [1, 4] (c) f (x) = x 3/4 on [0, 16] (e) f (x) = 25 x 2 on [ 3, 4] (b) f (x) = 3x 2 5x + 1 on [2, 5] x+3 on [1, 3] (d) f (x) = x 4 1 on [0, 2] (f ) f (x) = x 4 Draw Quick Response Code In Visual Studio .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Make QR Code In Visual Basic .NET Using Barcode generator for .NET Control to generate, create QR Code image in Visual Studio .NET applications. CHAP. 17] ANSI/AIM Code 128 Creation In .NET Framework Using Barcode maker for .NET framework Control to generate, create Code 128 image in .NET framework applications. Data Matrix ECC200 Drawer In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create Data Matrix image in Visual Studio .NET applications. THE MEANVALUE THEOREM AND THE SIGN OF THE DERIVATIVE
Print Code 3 Of 9 In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create Code 39 image in Visual Studio .NET applications. EAN8 Generation In .NET Framework Using Barcode creator for .NET Control to generate, create European Article Number 8 image in VS .NET applications. 17.12 Determine where the function f is increasing and where it is decreasing. Then sketch the graph of f . (a) f (x) = 3x + 1 (d) f (x) = 1 4x x 2 (g) f (x) = x 3 9x 2 + 15x 3 (b) f (x) = 2x + 2 (e) f (x) = 1 x2 1 (h) f (x) = x + x (c) f (x) = x 2 4x + 7 1 (f ) f (x) = 9 x2 3 (i) f (x) = x 3 12x + 20 Bar Code Creator In Java Using Barcode generator for Eclipse BIRT Control to generate, create bar code image in BIRT applications. Scanning Barcode In .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications. 17.13 Let f be a differentiable function such that f (x) = 0 for all x in the open interval (a, b). Prove that there is at most one zero of f (x) in (a, b). [Hint: Assume, for the sake of contradiction, that c and d are two zeros of f , with a < c < d < b, and apply Rolle s theorem on the interval [c, d].] 17.14 Consider the polynomial f (x) = 5x 3 2x 2 + 3x 4. (a) Show that f has a zero between 0 and 1. (b) Show that f has only one real zero. [Hint: Use Problem 17.13.] 17.15 Assume f continuous over [0, 1] and assume that f (0) = f (1). Which one(s) of the following assertions must be true (a) If f has an absolute maximum at c in (0, 1), then f (c) = 0. (b) f exists on (0, 1). (c) f (c) = 0 for some c in (0, 1). (d) lim f (x) = f (c) for all c in (0, 1). Code 39 Reader In .NET Framework Using Barcode decoder for .NET framework Control to read, scan read, scan image in .NET applications. Code 39 Extended Printer In Visual Studio .NET Using Barcode generation for Reporting Service Control to generate, create Code 3 of 9 image in Reporting Service applications. (e) f has an absolute maximum at some point c in (0, 1). 17.16 Let f and g be differentiable functions. (a) If f (a) = g(a) and f (b) = g(b), where a < b, show that f (c) = g (c) for some c in (a, b). (b) If f (a) g(a) and f (x) > g (x) for all x, show that f (x) > g(x) for all x > a. (c) If f (x) > g (x) for all x, show that the graphs of f and g intersect at most once. [Hint: In each part, apply the appropriate theorem to the function h(x) = f (x) g(x).] 17.17 Let f be a differentiable function on an open interval (a, b). (a) If f is increasing on (a, b), prove that f (x) 0 for every x in (a, b). f (x + h) f (x) and Problem 9.10(a) applies. Hint: f (x) = lim + h h 0 (b) If f is decreasing on (a, b), prove that f (x) 0 for every x in (a, b). 17.18 The meanvalue theorem predicts the existence of what point on the graph of y = 3 x between (27, 3) and (125, 5) Encoding Bar Code In Visual C# Using Barcode encoder for .NET framework Control to generate, create barcode image in .NET framework applications. Create Bar Code In Visual Basic .NET Using Barcode generation for .NET framework Control to generate, create barcode image in Visual Studio .NET applications. 17.19 (Generalized Rolle s Theorem) Assume f is continuous on [a, b] and differentiable on (a, b). If f (a) = f (b), prove that there is a point c in (a, b) such that f (c) = 0. [Hint: Apply Rolle s theorem to g(x) = f (x) f (a).] 17.20 Let f (x) = x 3 4x 2 + 4x and g(x) = 1 for all x. (a) Find the intersection of the graphs of f and g. (b) Find the zeros of f . (c) If the domain of f is restricted to the closed interval [0, 3], what would be the range of f 17.21 Prove that 8x 3 6x 2 2x + 1 has a zero between 0 and 1. [Hint: Apply Rolle s theorem to the function 2x 4 2x 3 x 2 + x.] 17.22 Show that x 3 + 2x 5 = 0 has exactly one real root. Code 3/9 Decoder In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. ECC200 Generator In .NET Framework Using Barcode creation for Reporting Service Control to generate, create Data Matrix 2d barcode image in Reporting Service applications. 
