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Question Number 30212 by abdo imad last updated on 18/Feb/18

$$letf[a,b]\to Rcontinueletsupposefderivableon[a,b]$$ $$and\mathrm{\forall }x\in [a,b]f\left(x\right)>0provethat$$ $$\mathrm{\exists }c\in ]a,b[/\frac{f\left(b\right)}{f\left(a\right)}=e^{(b-a)\frac{f^{,}\left(c\right)}{f\left(c\right)}}.$$

Commented byabdo imad last updated on 21/Feb/18

$$duetof\left(x\right)>0letput\phi \left(x\right)=ln\left(f\left(x\right)\right)\phi iscontinueon[a,b]$$ $$T.A.F\Rightarrow \mathrm{\exists }c\in ]a,b[/\phi \left(b\right)-\phi \left(a\right)=(b-a){\phi }^{{}^{\prime }}\left(c\right)\Rightarrow$$ $$ln\left(f\left(b\right)\right)-ln\left(f\left(a\right)\right)=(b-a)\frac{f^{{}^{\prime }}\left(c\right)}{f\left(c\right)}\Rightarrow ln\left(\frac{f\left(b\right)}{f\left(a\right)}\right)=(b-a)\frac{f^{{}^{\prime }}\left(c\right)}{f\left(c\right)}$$ $$\Rightarrow \mathrm{\exists }c\in ]a,b[/\frac{f\left(b\right)}{f\left(a\right)}=e^{(b-a)\frac{f^{{}^{\prime }}\left(c\right)}{f\left(c\right)}}.$$

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