If '(c) = 0 and (c) is neither a local min nor a local max, must x

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If ƒ'(c) = 0 and ƒ(c) is neither a local min nor a local max, must x = c be a point of inflection? This is true for “reasonable” functions (including the functions studied in this text), but it is not true in general. Let

f(x) = x sin / for x = 0 for x = 0

(a) Use the limit definition of the derivative to show that ƒ'(0) exists and ƒ'(0) = 0.
(b) Show that ƒ(0) is neither a local min nor a local max.
(c) Show that ƒ'(x) changes sign infinitely often near x = 0. Conclude that x = 0 is not a point of inflection.

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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