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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-

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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]and∀x∈[a,b]f(x)>0provethat∃c∈]a,b[/f(b)f(a)=e(b−a)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.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).

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