Let g R R be defined by g(x) x 2x2 sin(1 x) for x 0 and g(0) 0 Show that g(0) 1, but in every neighborhood of 0 the derivative g(x) takes on both positive and negative values Thus g is not monotonic in any neighborhood of 0

Question: Let g : R R be defined by g(x) := x + 2x2 sin(1/x) for x 0 and g(0) := 0. Show that

Let g : R → R be defined by g(x) := x + 2x2 sin(1/x) for x ≠ 0 and g(0) := 0. Show that gʹ(0) = 1, but in every neighborhood of 0 the derivative gʹ(x) takes on both positive and negative values. Thus g is not monotonic in any neighborhood of 0.

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