Suppose I have $f(x)A+g(x)B+h(x)C \ge 0$. Here $A,B,C$ can be positive or negative and $f,g,h$ are nonnegative. I would like to obtain a condition for $f,g,$ and $h$ such that $f'(x)A+g'(x)B+h'(x)C \ge 0$. I will appreciate any substantial comments.
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As the problem is currently stated, $A$, $B$, $C$, and $f$, $g$, $h$ are unfortunately not relevant. Let $$W(x)=Af(x)+Bg(x)+Ch(x).$$
The problem states that $W(x)\ge 0$, presumably for all $x$, and asks for conditions under which $W'(x) \ge 0$ for all $x$.
The condition $W(x) \ge 0$ cannot be of much help. We could ask for $W(x)$ to be non-decreasing, but that is really only a minor restatement of $W'(x)\ge 0$. Apart from that sort of thing, there is no nice condition on a general function that will ensure a non-negative derivative. And despite the apparent complexity of $W(x)$, there are no conditions on it apart from $W(x) \ge 0$.