Question Number 30212 by abdo imad last updated on 18/Feb/18
![let f [a,b]→R continue let suppose f derivable on[a,b] and ∀ x ∈[a,b] f(x)>0 prove that ∃c∈]a,b[ / ((f(b))/(f(a)))= e^((b−a)((f^, (c))/(f(c)))) .](https://www.tinkutara.com/question/Q30212.png)
$${let}\:{f}\:\:\left[{a},{b}\right]\rightarrow{R}\:{continue}\:{let}\:{suppose}\:{f}\:{derivable}\:{on}\left[{a},{b}\right] \\ $$$${and}\:\forall\:{x}\:\in\left[{a},{b}\right]\:\:{f}\left({x}\right)>\mathrm{0}\:{prove}\:{that} \\ $$$$\left.\exists{c}\in\right]{a},{b}\left[\:/\:\:\frac{{f}\left({b}\right)}{{f}\left({a}\right)}=\:{e}^{\left({b}−{a}\right)\frac{{f}^{,} \left({c}\right)}{{f}\left({c}\right)}} .\right. \\ $$
Commented by abdo imad last updated on 21/Feb/18
![due to f(x)>0 let put ϕ(x)=ln(f(x)) ϕ is continue on [a,b] T.A.F ⇒ ∃c∈]a,b[ /ϕ(b)−ϕ(a)=(b−a)ϕ^′ (c)⇒ ln(f(b))−ln(f(a))=(b−a)((f^′ (c))/(f(c))) ⇒ln(((f(b))/(f(a))))=(b−a)((f^′ (c))/(f(c))) ⇒∃c ∈]a,b[ / ((f(b))/(f(a)))=e^((b−a) ((f^′ (c))/(f(c)))) .](https://www.tinkutara.com/question/Q30392.png)
$${due}\:{to}\:\:{f}\left({x}\right)>\mathrm{0}\:\:{let}\:{put}\:\varphi\left({x}\right)={ln}\left({f}\left({x}\right)\right)\:\varphi\:{is}\:{continue}\:{on}\:\left[{a},{b}\right] \\ $$$$\left.{T}.{A}.{F}\:\Rightarrow\:\exists{c}\in\right]{a},{b}\left[\:\:/\varphi\left({b}\right)−\varphi\left({a}\right)=\left({b}−{a}\right)\varphi^{'} \left({c}\right)\Rightarrow\right. \\ $$$${ln}\left({f}\left({b}\right)\right)−{ln}\left({f}\left({a}\right)\right)=\left({b}−{a}\right)\frac{{f}^{'} \left({c}\right)}{{f}\left({c}\right)}\:\Rightarrow{ln}\left(\frac{{f}\left({b}\right)}{{f}\left({a}\right)}\right)=\left({b}−{a}\right)\frac{{f}^{'} \left({c}\right)}{{f}\left({c}\right)} \\ $$$$\left.\Rightarrow\exists{c}\:\in\right]{a},{b}\left[\:/\:\frac{{f}\left({b}\right)}{{f}\left({a}\right)}={e}^{\left({b}−{a}\right)\:\frac{{f}^{'} \left({c}\right)}{{f}\left({c}\right)}} \:\:.\right. \\ $$