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Question Number 30212 by abdo imad last updated on 18/Feb/18 | ||
$${let}\:{f}\:\:\left[{a},{b}\right]\rightarrow{R}\:{continue}\:{let}\:{suppose}\:{f}\:{derivable}\:{on}\left[{a},{b}\right] \\ $$ $${and}\:\forall\:{x}\:\in\left[{a},{b}\right]\:\:{f}\left({x}\right)>\mathrm{0}\:{prove}\:{that} \\ $$ $$\left.\exists{c}\in\right]{a},{b}\left[\:/\:\:\frac{{f}\left({b}\right)}{{f}\left({a}\right)}=\:{e}^{\left({b}−{a}\right)\frac{{f}^{,} \left({c}\right)}{{f}\left({c}\right)}} .\right. \\ $$ | ||
Commented byabdo imad last updated on 21/Feb/18 | ||
$${due}\:{to}\:\:{f}\left({x}\right)>\mathrm{0}\:\:{let}\:{put}\:\varphi\left({x}\right)={ln}\left({f}\left({x}\right)\right)\:\varphi\:{is}\:{continue}\:{on}\:\left[{a},{b}\right] \\ $$ $$\left.{T}.{A}.{F}\:\Rightarrow\:\exists{c}\in\right]{a},{b}\left[\:\:/\varphi\left({b}\right)−\varphi\left({a}\right)=\left({b}−{a}\right)\varphi^{'} \left({c}\right)\Rightarrow\right. \\ $$ $${ln}\left({f}\left({b}\right)\right)−{ln}\left({f}\left({a}\right)\right)=\left({b}−{a}\right)\frac{{f}^{'} \left({c}\right)}{{f}\left({c}\right)}\:\Rightarrow{ln}\left(\frac{{f}\left({b}\right)}{{f}\left({a}\right)}\right)=\left({b}−{a}\right)\frac{{f}^{'} \left({c}\right)}{{f}\left({c}\right)} \\ $$ $$\left.\Rightarrow\exists{c}\:\in\right]{a},{b}\left[\:/\:\frac{{f}\left({b}\right)}{{f}\left({a}\right)}={e}^{\left({b}−{a}\right)\:\frac{{f}^{'} \left({c}\right)}{{f}\left({c}\right)}} \:\:.\right. \\ $$ | ||