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APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING in .NET framework
APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING Denso QR Bar Code Decoder In VS .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Create QR Code In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create QR Code image in VS .NET applications. [CHAP. 23
Quick Response Code Decoder In .NET Using Barcode recognizer for .NET framework Control to read, scan read, scan image in .NET framework applications. Barcode Creator In Visual Studio .NET Using Barcode creator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Fig. 2314 Bar Code Reader In .NET Framework Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET applications. QRCode Printer In C#.NET Using Barcode creator for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications. 23.12 Sketch the graph of a continuous function f such that: (a) f (1) = 2, f (1) = 0, f (x) > 0 for all x (b) f (2) = 3, f (2) = 0, f (x) < 0 for all x (c) f (1) = 1, f (x) < 0 for x > 1, f (x) > 0 for x < 1, lim f (x) = + , lim f (x) = QR Printer In VS .NET Using Barcode generator for ASP.NET Control to generate, create QR Code image in ASP.NET applications. Print Quick Response Code In Visual Basic .NET Using Barcode generator for .NET framework Control to generate, create Quick Response Code image in Visual Studio .NET applications. x + x + x
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Encoding Bar Code In VS .NET Using Barcode maker for ASP.NET Control to generate, create barcode image in ASP.NET applications. Creating Data Matrix In Java Using Barcode creation for Java Control to generate, create Data Matrix image in Java applications. (f) f (0) = 0, f (x) > 0 for x < 0, f (x) < 0 for x > 0, lim f (x) = + , lim f (x) = +
Scan Bar Code In VS .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET framework applications. Barcode Printer In Java Using Barcode creation for Java Control to generate, create bar code image in Java applications. x 0 x 0+
(g) f (0) = 1, f (x) < 0 if x = 0, lim f (x) = 0, lim f (x) = x 0+ x 0
23.13 Let f (x) = xx 1 for x in [ 1, 2]. (a) At what values of x is f continuous (b) At what values of x is f differentiable Calculate f (x). [Hint: Distinguish the cases x > 1 and x < 1.] (c) Where is f an increasing function (d) Calculate f (x). (e) Where is the graph of f concave upward, and where concave downward (f ) Sketch the graph of f . 23.14 Given functions f and g such that, for all x, (i) (g(x))2 (f (x))2 = 1; (ii) f (x) = (g(x))2 ; (iii) f (x) and g (x) exist; (iv) g(x) < 0; (v) f (0) = 0. Show that: (a) g (x) = f (x)g(x); (b) g has a relative maximum at x = 0; (c) f has a point of in ection at x = 0. 23.15 For what value of k will x kx 1 have a relative maximum at x = 2 23.16 Let f (x) = x 4 + Ax 3 + Bx 2 + Cx + D. Assume that the graph of y = f (x) is symmetric with respect to the yaxis, has a relative maximum at (0, 1), and has an absolute minimum at (k, 3). Find A, B, C, and D, as well as the possible value(s) of k. 23.17 Prove Theorem 23.1. [Hint: Assume that f (x) > 0 on (a, b), and let c be in (a, b). The equation of the tangent line at x = c is y = f (c)(x c) + f (c). It must be shown that f (x) > f (c)(x c) + f (c). But the meanvalue theorem gives f (x) = f (x )(x c) + f (c) where x is between x and c, and since f (x) > 0 on (a, b) , f is increasing.] 23.18 Give a rigorous proof of the secondderivative test (Theorem 23.3). [Hint: Assume f (c) = 0 and f (c) < 0. Since f (c) < 0, f (c + h) f (c) f (c + h) f (c) < 0. So, there exists > 0 such that, for h < , < 0, and since f (c) = 0, lim h h h 0 f (c + h) f (c) = f (c + h1 ) f (c + h) < 0 for h > 0 and f (c + h) > 0 for h < 0. By the meanvalue theorem, if h < , h CHAP. 23] APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
for some c + h1 between c and c + h. So, h1  < h, and whether h > 0 or h < 0, we can deduce that f (c + h) f (c) < 0; that is, f (c + h) < f (c) . Thus, f has a relative maximum at c. The case when f (c) > 0 is reduced to the rst case by considering f .] 23.19 Consider f (x) = 3(x 2 1) . x2 + 3 (a) Find all open intervals where f is increasing. (b) Find all critical points and determine whether they correspond to relative maxima, relative minima, or neither. (c) Describe the concavity of the graph of f and nd all in ection points (if any). (d) Sketch the graph of f . Show any horizontal or vertical asymptotes. 23.20 In the graph of y = f (x) in Fig. 2315: (a) nd all x such that f (x) > 0; (b) nd all x such that f (x) > 0. Fig. 2315

