Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows
$$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$
where $0<C_i<1$ and $0\leq D_i<2\pi$ are real-valued constants and $x\in \mathbb R$. Now we restrict $x$ to an interval $]a,b[$, s.t.,
\begin{align}\forall i, j \in \{1, \ldots, n\} ~~\forall x\!\in]a, b[:& ~~f_i(x)<f_j(x) &~~\lor \forall x\!\in]a, b[:& ~~f_i(x)>f_j(x) \\\forall i, j \in \{1, \ldots, n\} ~~\forall x\!\in]a, b[:& ~~f_i'(x)<f_j'(x) &~~\lor \forall x\!\in]a, b[:& ~~f_i'(x)>f_j'(x) \\\forall i, j \in \{1, \ldots, n\} ~~\forall x\!\in]a, b[:& ~~f_i''(x)<f_j''(x) &~~\lor \forall x\!\in]a, b[:& ~~f_i''(x)>f_j''(x) \\\forall i \in \{1, \ldots, n\} ~~\forall x\!\in]a, b[:& ~~f_i(x)<0 &~~\lor \forall x\!\in]a, b[:& ~~f_i(x)>0 \\\forall i \in \{1, \ldots, n\} ~~\forall x\!\in]a, b[:& ~~f_i'(x)<0 &~~\lor \forall x\!\in]a, b[:& ~~f_i'(x)>0 \\\forall i \in \{1, \ldots, n\} ~~\forall x\!\in]a, b[:& ~~f_i''(x)<0 &~~\lor \forall x\!\in]a, b[:& ~~f_i''(x)>0\end{align}
In words within the interval $]a,b[$ the following holds:
- there is a strict total order over all functions
- there is a strict total order of their first derivative (although it does not need to be the same order)
- there is a strict total order of their second derivative (although it does not need to be the same order)
- no function intersects the $x$-axis
- no function has an extremum
- no function has an inflection point
Question: How many minima can the function $f_{\Sigma}(x) = \sum \limits_{i=1}^{n} f_i(x)$ have within the interval $]a,b[$?
My hope is that the answer to the question is that there can only be one or a constant number of minima, but I would also be happy with any polynomial upper OR exponential lower bound.
I have tried to accumulate the functions into a constant number of functions by grouping them into four sets $F^+_+, F^+_-, F^-_+$ and $F^-_-$, s.t., any function in the set $F^+_-$ has positive first derivative and negative second derivative (and similar for the other three sets). Summing up all of the functions in these sets yields four functions $$f^+_+(x) =\!\! \sum\limits_{f\in F^+_+} f(x),~~~f^+_-(x) =\!\! \sum\limits_{f\in F^+_-} f(x),~~~f^-_+(x) =\!\! \sum\limits_{f\in F^-_+} f(x),~~~f^-_-(x) =\!\! \sum\limits_{f\in F^-_-} f(x)$$and we can observe that we retain that for, e.g., $f^+_-$ the first derivative is entirely positive and the second is entirely negative within $]a,b[$ (and similar for $f^+_+, f^-_-, f^-_+$). Note that these function do not necessarily have a total order anymore, nor do their derivatives. The hope was that reducing the number of functions to just 4 would be helpful, but I could not figure out how to proceed from here.
I have also tried to plot these functions in Mathematica and I have not encountered any instance with more than one minimum, which is where my suspicion comes from that the answer is 1.
Alternative follow up question: If it is for some reason not possible to give any upper bound on the number of minima, what would be a sufficient property of these functions to obtain one?