Assume that exists and let c be a point of inflection of . (a) Use the

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Assume that ƒ" exists and let c be a point of inflection of ƒ.

(a) Use the method of Exercise 62 to prove that the tangent line at x = c crosses the graph (Figure 21). Show that G(x) changes sign at x = c.

image


Data From Exercise 62

Prove that if ƒ" exists and ƒ"(x) > 0 for all x, then the graph of ƒ“sits above” its tangent lines.

(a) For any c, set G(x) = f(x) = f'(c)(x - c) - f(c). It is sufficient to prove that G(x) > 0 for all c.

(b) Verify this conclusion for ƒ(x) = x / 3x2 + 1 by graphing ƒ and the tangent line at each inflection point on the same set of axes.

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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