关于数学分析的数学竞赛题

$f命题：$设$fleft( x ight) in {C^1}left( {0, + infty } ight),fleft( x ight) > 0left( {0 < x <  + infty } ight)$，证明：$mathop {underline {lim } }limits_{x o egin{array}{*{20}{c}}{ + infty } end{array}} frac{{f'left( x ight)}}{{{f^p}left( x ight)}} leqslant 0left( {p > 1} ight)$

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$f命题：$