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barcode print in asp net Fig. 231 Concavity upward in VS .NET
Fig. 231 Concavity upward QR Code JIS X 0510 Reader In .NET Framework Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in VS .NET applications. Encode QR In .NET Using Barcode drawer for .NET Control to generate, create QR image in Visual Studio .NET applications. A curve is said to be concave downward if it has the shape of a cap or part of a cap (see Fig. 232). In mathematical terms, a curve is concave downward if it lies below the tangent line at an arbitrary point of the curve [see Fig. 232(a)]. Decoding Denso QR Bar Code In .NET Using Barcode decoder for .NET Control to read, scan read, scan image in .NET applications. Barcode Drawer In .NET Framework Using Barcode drawer for Visual Studio .NET Control to generate, create barcode image in VS .NET applications. Copyright 2008, 1997, 1985 by The McGrawHill Companies, Inc. Click here for terms of use.
Bar Code Reader In .NET Framework Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. QR Code 2d Barcode Creation In C# Using Barcode printer for VS .NET Control to generate, create Quick Response Code image in .NET framework applications. CHAP. 23] QR Code Maker In VS .NET Using Barcode maker for ASP.NET Control to generate, create QR Code ISO/IEC18004 image in ASP.NET applications. QR Code Encoder In Visual Basic .NET Using Barcode maker for VS .NET Control to generate, create QR image in .NET applications. APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
UCC.EAN  128 Generator In VS .NET Using Barcode generator for .NET Control to generate, create GTIN  128 image in Visual Studio .NET applications. Make Bar Code In VS .NET Using Barcode drawer for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Fig. 232 Concavity downward A curve may, of course, be composed of parts of different concavity. The curve in Fig. 233 is concave downward from A to B, concave upward from B to C, concave downward from C to D, and concave upward from D to E. A point on the curve at which the concavity changes is called an in ection point. B, C, and D are in ection points in Fig. 233. GS1 DataBar14 Creator In .NET Using Barcode encoder for .NET framework Control to generate, create GS1 DataBar Truncated image in VS .NET applications. Identcode Generation In Visual Studio .NET Using Barcode encoder for Visual Studio .NET Control to generate, create Identcode image in Visual Studio .NET applications. Fig. 233 From Fig. 231 we see that if we move from left to right along a curve that is concave upward, the slope of the tangent line increases. The slope either becomes less negative or more positive. Conversely, if the tangent line has this property, the curve must be concave upward. Now for a curve y = f (x), the tangent line will certainly have this property if f (x) > 0 since, in that case, Theorem 17.3 implies that the slope f (x) of the tangent line will be an increasing function. By a similar argument, we see that if f (x) < 0, the slope of the tangent line is decreasing, and from Fig. 232 we see that the curve y = f (x) is concave downward. This yields: Theorem 23.1: If f (x) > 0 for all x in (a, b), then the graph of f is concave upward between x = a and x = b. If f (x) < 0 for all x in (a, b), then the graph of f is concave downward between x = a and x = b. For a rigorous proof of Theorem 23.1, see Problem 23.17. Corollary 23.2: If the graph of f has an in ection point at x = c, and f exists and is continuous at x = c, then f (c) = 0. In fact, if f (c) = 0, then f (c) > 0 or f (c) < 0. If f (c) > 0, then f (x) > 0 for all x in some open interval containing c, and the graph would be concave upward in that interval, contradicting the assumption that there is an in ection point at x = c. We get a similar contradiction if f (c) < 0, for in that case, the graph would be concave downward in an open interval containing c. EXAMPLES Painting Data Matrix In Java Using Barcode encoder for Eclipse BIRT Control to generate, create Data Matrix ECC200 image in BIRT reports applications. Bar Code Decoder In Visual Basic .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. (a) Consider the graph of y = x 3 [see Fig. 234(a)]. Here y = 3x 2 and y = 6x. Since y > 0 when x > 0, and y < 0 when x < 0, the curve is concave upward when x > 0, and concave downward when x < 0. There is an in ection point at the origin, where the concavity changes. This is the only possible in ection point, for if y = 6x = 0, then x must be 0. Encode Barcode In None Using Barcode generator for Word Control to generate, create bar code image in Office Word applications. Barcode Creator In None Using Barcode generation for Online Control to generate, create barcode image in Online applications. APPLICATIONS OF THE SECOND DERIVATIVE AND GRAPH SKETCHING
Matrix 2D Barcode Drawer In Java Using Barcode creator for Java Control to generate, create 2D Barcode image in Java applications. Code 39 Extended Encoder In None Using Barcode drawer for Font Control to generate, create USS Code 39 image in Font applications. [CHAP. 23
1D Generation In Java Using Barcode maker for Java Control to generate, create 1D image in Java applications. EAN13 Recognizer In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. (b) If f (c) = 0, the graph of f need not have an in ection point at x = c. For instance, the graph of f (x) = x 4 [see Fig. 234(b)] has a relative minimum, not an in ection point, at x = 0, where f (x) = 12x 2 = 0. Fig. 234 23.2 TEST FOR RELATIVE EXTREMA We already know, from 14, that the condition f (c) = 0 is necessary, but not suf cient, for a differentiable function f to have a relative maximum or minimum at x = c. We need some additional information that will tell us whether a function actually has a relative extremum at a point where its derivative is zero. Theorem 23.3 (SecondDerivative Test for Relative Extrema): If f (c) = 0 and f (c) < 0, then f has a relative maximum at c. If f (c) = 0 and f (c) > 0, then f has a relative minimum at c. Proof: If f (c) = 0, the tangent line to the graph of f is horizontal at x = c. If, in addition, f (c) < 0, then, by Theorem 23.1,1 the graph of f is concave downward near x = c. Hence, near x = c, the graph of f must lie below the horizontal line through (c, f (c)); f thus has a relative maximum at x = c [see Fig. 235(a)]. A similar argument leads to a relative minimum when f (c) > 0 [see Fig. 235(b)]. Fig. 235

