 # Differentiable Function Problems

## Transcribed Text

Explain the following example so it can be used in a presentation: 5. A differentiable function whose derivative is positive at a point but which is not monotonic in any neighborhood of the point. The function f(x) = ( x + 2x 2 sin 1 x , if x 6= 0, 0, if x = 0 has the derivative f 0 (x) = ( 1 + 4x sin 1 x − 2 cos 1 x , if x 6= 0, 1, if x = 0 In every neighborhood of 0 the function f 0 (x) has both positive and negative values.

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