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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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    Unless you tell us more about $f$, $g$, $h$ and the $x$-domain $I$ where all this should hold the condition $f'(x)A+g'(x)B+h'(x)C \ge 0$ cannot be transformed into something simpler. If $I$ is a compact interval then the assumption $f(x)A+g(x)B+h(x)C \ge 0$ is useless, because by adding a suitable constant to $f$ you always can force it to hold without changing the derivative(s).2011-07-09
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    $I$ is a compact interval, and $A, B, C $are given.2011-07-09

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