关于数学分析的数学竞赛题

$\bf命题：$设$f\left( x \right) \in {C^1}\left( {0, + \infty } \right),f\left( x \right) > 0\left( {0 < x <  + \infty } \right)$，证明：$\mathop {\underline {\lim } }\limits_{x \to \begin{array}{*{20}{c}}{ + \infty } \end{array}} \frac{{f'\left( x \right)}}{{{f^p}\left( x \right)}} \leqslant 0\left( {p > 1} \right)$

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$\bf命题：$