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.
Conditions for some inequality
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inequality
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0$I$ is a compact interval, and $A, B, C $are given. – 2011-07-09
1 Answers
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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 $W(x) \ge 0$.
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0I am not asking a general condition. I am asking conditions given A, B, and$C$fixed. – 2011-07-09