 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
More Maximum and Minimum Problems in .NET
More Maximum and Minimum Problems Scanning QR In Visual Studio .NET Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in VS .NET applications. Drawing QR Code 2d Barcode In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications. Until now we have been able to nd the absolute maxima and minima of differentiable functions only on closed intervals (see Section 14.2). The following result often enables us to handle cases where the function is de ned on a halfopen interval, open interval, in nite interval, or the set of all real numbers. Remember that, in general, there is no guarantee that a function has an absolute maximum or an absolute minimum on such domains. Theorem 24.1: Let f be a continuous function on an interval I , with a single relative extremum within I . Then this relative extremum is also an absolute extremum on I . Intuitive Argument: Refer to Fig. 241. Suppose that f has a relative maximum at c and no other relative extremum inside I . Take any other number d in I . The curve moves downward on both sides of c. Hence, if the value f (d) were greater than f (c), then, at some point u between c and d, the curve would have to change direction and start moving upward. But then f would have a relative minimum at u, contradicting our assumption. The result for a relative minimum follows by applying to f the result just obtained for a relative maximum. Decode QR Code JIS X 0510 In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. Encode Bar Code In VS .NET Using Barcode generation for VS .NET Control to generate, create bar code image in .NET applications. For a rigorous proof, see Problem 24.20.
Barcode Decoder In VS .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Painting QR In Visual C#.NET Using Barcode generator for .NET Control to generate, create QRCode image in .NET framework applications. Fig. 241 Paint QR Code In VS .NET Using Barcode creation for ASP.NET Control to generate, create QRCode image in ASP.NET applications. Making QRCode In VB.NET Using Barcode generator for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. Copyright 2008, 1997, 1985 by The McGrawHill Companies, Inc. Click here for terms of use.
Make Barcode In .NET Using Barcode encoder for Visual Studio .NET Control to generate, create barcode image in .NET framework applications. Encode Code 128 Code Set B In VS .NET Using Barcode creation for .NET Control to generate, create ANSI/AIM Code 128 image in .NET applications. CHAP. 24] Data Matrix ECC200 Creation In Visual Studio .NET Using Barcode encoder for .NET Control to generate, create ECC200 image in VS .NET applications. MSI Plessey Creation In VS .NET Using Barcode printer for VS .NET Control to generate, create MSI Plessey image in Visual Studio .NET applications. MORE MAXIMUM AND MINIMUM PROBLEMS
Printing Bar Code In Java Using Barcode generation for Android Control to generate, create barcode image in Android applications. Code 39 Recognizer In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. EXAMPLES
Making Barcode In Java Using Barcode drawer for Java Control to generate, create bar code image in Java applications. EAN13 Reader In C# Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications. (a) Find the shortest distance from the point P(1, 0) to the parabola x = y2 [see Fig. 242(a)]. 1D Barcode Maker In Visual Studio .NET Using Barcode generator for ASP.NET Control to generate, create 1D Barcode image in ASP.NET applications. Scanning Barcode In C#.NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Fig. 242 European Article Number 13 Creation In None Using Barcode creator for Microsoft Word Control to generate, create European Article Number 13 image in Word applications. Reading Code 39 In .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in VS .NET applications. The distance from an arbitrary point Q(x, y) on the parabola to the point P(1,0) is, by (2.1), u= = = (x 1)2 + y2 (x 1)2 + x [y2 = x at Q] x2 x + 1 x 2 2x + 1 + x =
But minimizing u is equivalent to minimizing u2 F(x) = x 2 x + 1 on the interval [0, + ) (the value of x is restricted by the fact that x = y2 0). F (x) = 2x 1 The only critical number is the solution of F (x) = 2x 1 = 0 or x= 1 2 F (x) = 2 Now F ( 1 ) = 2 > 0. So by the secondderivative test, the function F has a relative minimum at x = 1 , Theorem 24.1 implies 2 2 that this is an absolute minimum. When x = 1 , 2 1 y2 = x = 2 and 1 2 2 1 y = = = 2 2 2 2 2/2) and ( 1 , 2/2). 2 Thus, the points on the parabola closest to (1,0) are ( 1 , 2 (b) An open box (that is, a box without a top) is to be constructed with a square base [see Fig. 242(b)] and is required to have a volume of 48 cubic inches. The bottom of the box costs 3 cents per square inch, whereas the sides cost 2 cents per square inch. Find the dimensions that will minimize the cost of the box. Let x be the side of the square bottom, and let h be the height. Then the cost of the bottom is 3x 2 and the cost of each of the four sides is 2xh, giving a total cost of C = 3x 2 + 4(2xh) = 3x 2 + 8xh The volume is V = 48 = x 2 h. Hence, h = 48/x 2 and C = 3x 2 + 8x 48 x2 = 3x 2 + 384 = 3x 2 + 384x 1 x MORE MAXIMUM AND MINIMUM PROBLEMS
[CHAP. 24
which is to be minimized on (0, + ). Now 384 dC = 6x 384x 2 = 6x 2 dx x and so the critical numbers are the solutions of 6x 384 =0 x2 384 6x = 2 x x 3 = 64 x=4 Now 768 d2C = 6 ( 2)384x 3 = 6 + 3 > 0 dx 2 x for all positive x; in particular, for x = 4. By the secondderivative test, C has a relative minimum at x = 4. But since 4 is the only positive critical number and C is continuous on (0, + ), Theorem 24.1 tells us that C has an absolute minimum at x = 4. When x = 4, 48 48 h= 2 = =3 16 x So, the side of the base should be 4 inches and the height 3 inches.

