I have a function $g(x)$ of $0\leq x \leq 1$, as the sum of the functions in the following forms
\begin{align} f_1(x)&=(2-x)\frac{1-n}{m_1-n},\\ f_2(x)&=(2-x)x\frac{1-xn}{m_2-xn},\\ f_3(x)&=(2-x)(1-x)\frac{1-(1-x)n}{m_3-(1-x)n} \end{align}
where $m_1,m_2,m_3,n$ have the same domain as $x$ with the constraints that all the denominators should be positive. So, it is clear that all the functions have positive value.
I want to find the minimum of $g(x)$. I want to prove that $g(x)$ is an increasing or decreasing or unimodal (or even bimodal with single minimum) function, so I can use some search algorithms. I can prove that $f_1(x)$ and $f_3(x)$ are decreasing functions and $f_2(x)$ is an increasing function, based on their derivatives. So $g(x)$ is the sum of increasing and decreasing functions of $x$. Is that enough to prove that $g(x)$ has a single minimum?
Thanks in Advance.