The aim of this note is to present some results associated to the l’H\^{o}pital-type rules for the monotonicity (LMR). We will prove the monotonicity of the ratio \frac{f\left( x\right) -f\left( c\right) }{g\left( x\right)-g\left( c\right) }, were c is an arbitrary point from the interval \left( a,b\right). Also, we extend LMR to the ratio of two higher-order differentiable functions. We complete with some applications of our results.



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Marinescu, Dan Ștefan, Monea, Mihai