Can we predict the behavior of a linear composition of known functions?

Question

Let $$f(x) = a_1f_1(x)+...+a_Nf_N(x)$$ be a function composed of known continuous and smooth functions $f_1...f_N$ and $a_1...a_N$ some constants. the term "critical point" is defined as the roots of derivative. But in here, let us define the vector of critical points as: $$x_c = [x_{c1},x_{c2},...,x_{cK}]$$ which contains all zeros of the functions $f_1...f_N$ and their derivatives, sorted in ascending order: $x_{c1}<x_{c2}<...<x_{cK}$; that is, the points $x_c$ are either roots or the local minima/maxima of the consisting functions.

I want to isolate the intervals that contain one and only one critical point of $f(x)$. Is it possible if $x_c$ are at hand? Also is it correct to say that in an arbitrary interval of $[x_{ci},x_{ci+1}]$, there exists only one critical point?

Answer 1

No; there doesn't need to be any simple relationship between the roots/critical points of $f(x)$ and the roots/critical points of the $f_i$. Consider \begin{align*} f_1(x) &= \sin(x) + 2x\\ f_2(x) &= \sin(x) - 2x\\ \alpha_1 = \alpha_2 &= \frac{1}{2}. \end{align*} Then $x_c = \{0\}$ but of course $f(x)$ has tons of critical points (and roots).

Answer 2

As exemplified by @user7530 you need further assumptions before progress can be made. In the extreme case of $f_i : \mathbb{R} \rightarrow \mathbb{R}$ given by $f_i(x) = \exp(x)$ for $i=1,2$ and $a_1 + a_2 = 0$ you have $f(x) = f'(x) = 0$ for all $x \in \mathbb{R}$, while $f_i = f'_i$ has no zeros at all.