I am wondering how one can study a function whose roots exist only in complex plane $\mathbb{C}$.
A similar question have been asked here, but it is quite different from this one.
For example's sake let:
$$f(x) = \frac{e^x-e^{-x}}{2} $$
Now let's take a look at its first derivative:
$$f'(x)=\frac{e^x+e^{-x}}{2} =0 \quad (1)$$
Apparently there are only complex roots for $(1)$: $\:x=\frac{1}{2}i\left(2\pi n+\pi \right), n\in \mathbb{Z}$
(Similarly for the second derivative -in order to study the curvarture.)
The graph suggests that the curve $f(x)$ has:
- One saddle point at $x_0 = 0$.
- The function is increasing $\forall x \in D(f)$ (except, of course, the saddle point)
- I has an inflection point at $x_0 = 0$ and it is concave down and up respectively.
How can these facts be stated algebraically?
The first derivative is used to discuss monotonicity of the function and find stationary points (local extrema or saddle point). What matters is the sign of the first derivative (and the zeroes).
The second derivative is used to qualify the extrema and detect the inflection points. What matters is the sign of the second derivative (and the zeroes).
When you study the function on the real axis, what happens in the complex plane is completely irrelevant. In fact, it is not deemed to exist.
E.g.,
$$\frac{x^3}3+x$$ has no stationary point because $$x^2+1$$ remains strictly positive.