Question

Give an example of function that is differentiable, its derivative is positive at some point P but there is no neighboorhood of P such that function is monotone there.

Answer 1

Let f(x) = x + 2 * x^2 * sin (1/x) when x is not equal to 0, f(x) = 0 when x = 0.

Then the derivative:
f'(x) = 1 + 4x * sin (1/x) - 2cos (1/x) when x is not equal to 0 and
f'(0) = 1.

For any neighbourhood of x=0 the derivative f'(x) has positive and negative values.
So, there is no neighbourhood of x=0 such that f(x) is monotone there.