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Supplementary Problems in .NET
Supplementary Problems Denso QR Bar Code Reader In VS .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in VS .NET applications. QRCode Drawer In Visual Studio .NET Using Barcode generator for .NET Control to generate, create QR Code image in Visual Studio .NET applications. 24.4 A rectangular eld is to be fenced in so that the resulting area is 100 square meters. Find the dimensions of that eld (if any) for which the perimeter is: (a) a maximum; (b) a minimum. 24.5 Find the point(s) on the parabola 2x = y2 closest to the point (1, 0). 24.6 Find the point(s) on the hyperbola x 2 y2 = 2 closest to the point (0, 1). 24.7 A closed box with a square base is to contain 252 cubic feet. The bottom costs $5 per square foot, the top costs $2 per square foot, and the sides cost $3 per square foot. Find the dimensions that will minimize the cost. 24.8 Find the absolute maxima and minima (if any) of f (x) = x2 + 4 on the interval [0, 2). x 2 Recognize QR Code JIS X 0510 In .NET Framework Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Make Bar Code In VS .NET Using Barcode creator for .NET framework Control to generate, create barcode image in .NET framework applications. 24.9 A printed page must contain 60 square centimeters of printed material. There are to be margins of 5 centimeters on either side, and margins of 3 centimeters each on the top and the bottom. How long should the printed lines be in order to minimize the amount of paper used 24.10 A farmer wishes to fence in a rectangular eld of 10 000 square feet. The north south fences will cost $1.50 per foot, whereas the east west fences will cost $6.00 per foot. Find the dimensions of the eld that will minimize the cost. Reading Barcode In Visual Studio .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications. QR Code 2d Barcode Printer In Visual C#.NET Using Barcode generation for Visual Studio .NET Control to generate, create QRCode image in .NET framework applications. 1 4x 2 3 = 4(x + 1) + as x 1 . x 1 x 1
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Barcode Printer In C#.NET Using Barcode drawer for .NET framework Control to generate, create bar code image in .NET framework applications. Barcode Maker In Visual Studio .NET Using Barcode generation for Reporting Service Control to generate, create barcode image in Reporting Service applications. (b) Find the point on the graph where the tangent line has the greatest slope. 24.12 (a) Find the dimensions of the closed cylindrical can [see Fig. 244(a)] that will have a capacity of k volume units and use the minimum amount of material. Find the ratio of the height h to the radius r of the top and bottom. (The volume is V = r 2 h, and the lateral surface area is S = 2 rh.) (b) If the bottom and the top of the can have to be cut from square pieces of metal and the rest of these squares is wasted [see Fig. 244(b)], nd the dimensions that will minimize the amount of material used, and nd the ratio of the height to the radius. Fig. 244 Fig. 245 24.13 A thinwalled coneshaped cup is to hold 36 cubic inches of water when full. What dimensions will minimize the amount of material needed for the cup (The volume is V = 1 r 2 h and the surface area is A = rs; see Fig. 245.) 3 24.14 (a) Find the absolute extrema on [0, + ) (if they exist) of f (x) = x . (b) Sketch the graph of f . (x 2 + 1)3/2 24.15 A rectangular bin, open at the top, is required to contain a volume of 128 cubic meters. If the bottom is to be a square, at a cost of $2 per square meter, whereas the sides cost $0.50 per square meter, what dimensions will minimize the cost 24.16 The selling price P of an item is 100 0.02x dollars, where x is the number of items produced per day. If the cost C of producing and selling x items is 40x + 15 000 dollars per day, how many items should be produced and sold every day in order to maximize the pro t 24.17 Consider all lines through the point (1, 3) and intersecting the positive xaxis at A(x, 0) and the positive yaxis at B(0, y) (see Fig. 246). Find the line that makes the area of BOA a minimum. MORE MAXIMUM AND MINIMUM PROBLEMS
[CHAP. 24
Fig. 246 k 24.18 Consider the function f (x) = 1 x 2 + . (a) For what value of k will f have a relative minimum at x = 2 (b) For the value 2 x of k found in part (a), sketch the graph of f . (c) For what value(s) of k will f have an absolute minimum 24.19 Find the point(s) on the graph of 3x 2 + 10xy + 3y2 = 9 closest to the origin. [Hint: Minimize x 2 + y2 , making use of implicit differentiation.] 24.20 Fill in the gaps in the following proof of Theorem 24.1. Assume that f is continuous on an interval I and that f has a relative maximum on I at c. Assume, to the contrary, that d = c is a point in I with f (d) > f (c). On the closed interval I with endpoints c and d, f has an absolute minimum at some point u. Clearly, u = d. Also, u cannot lie in the interior of I . (Otherwise, f would have a relative minimum on I at u, contradicting the hypothesis that c is the only point in I at which f has a relative extremum.) Hence, u = c. This would imply that f (x) = f (c) in a subinterval of I and that f would have relative extrema at all points of that subinterval. This would contradict the hypothesis that c is the unique point in I at which f has a relative extremum. (A similar argument would show that f has an absolute minimum at c when it is assumed that f has a relative minimum at c.) 24.21 Prove the following theorem, similar to Theorem 24.1: If the graph of f is concave upward (downward) over an interval I , then any relative minimum (maximum) of f in I is an absolute minimum (maximum) on I . [Hint: Consider the relationship of the graph of f to the tangent line at the relative extremum.) 24.22 Find the absolute extrema (if any) of f (x) = x 2/5 1 x 7/5 on ( 1, 1]. 7

