barcode print in asp net Fig. 23-1 Concavity upward in VS .NET

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Fig. 23-1 Concavity upward
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A curve is said to be concave downward if it has the shape of a cap or part of a cap (see Fig. 23-2). In mathematical terms, a curve is concave downward if it lies below the tangent line at an arbitrary point of the curve [see Fig. 23-2(a)].
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Fig. 23-2 Concavity downward A curve may, of course, be composed of parts of different concavity. The curve in Fig. 23-3 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. 23-3.
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Fig. 23-3 From Fig. 23-1 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. 23-2 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
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(a) Consider the graph of y = x 3 [see Fig. 23-4(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.
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(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. 23-4(b)] has a relative minimum, not an in ection point, at x = 0, where f (x) = 12x 2 = 0.
Fig. 23-4
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 (Second-Derivative 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. 23-5(a)]. A similar argument leads to a relative minimum when f (c) > 0 [see Fig. 23-5(b)].
Fig. 23-5
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