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let-f-a-b-R-continue-let-suppose-f-derivable-on-a-b-and-x-a-b-f-x-gt-0-prove-that-c-a-b-f-b-f-a-e-b-a-f-c-f-c-




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)))) .
letf[a,b]Rcontinueletsupposefderivableon[a,b]andx[a,b]f(x)>0provethatc]a,b[/f(b)f(a)=e(ba)f,(c)f(c).
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))))   .
duetof(x)>0letputφ(x)=ln(f(x))φiscontinueon[a,b]T.A.Fc]a,b[/φ(b)φ(a)=(ba)φ(c)ln(f(b))ln(f(a))=(ba)f(c)f(c)ln(f(b)f(a))=(ba)f(c)f(c)c]a,b[/f(b)f(a)=e(ba)f(c)f(c).

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