Can linearization of a function around $x=0$ show whether first derivative is positive or negative?

Question

As title says, can linearization of a function $f(x)$ (by the method of taylor series around $x=0$) show whether first derivative of the function ($df/dx$) is positive or negative at $x=0$?

And.. the answer is obvious, meh.

Answer 1

Yes. The taylor series of $f$ centered at zero begins: $$f(0) + f'(0)x + \cdots$$ so if you know the taylor series at zero then you know the first derivative at zero.