O problema clasica de medie
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Cezar Lupu
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O problema clasica de medie
Fie \( f:[0,1]\to\mathbb{R} \) continua astfel incat \( \int_0^1 f(x)dx=0 \). Sa se arate ca exista \( c\in (0,1) \) astfel incat \( f(c)=\int_0^c f(x)dx \).
Last edited by Cezar Lupu on Wed Sep 26, 2007 10:37 pm, edited 1 time in total.
An infinite number of mathematicians walk into a bar. The first one orders a beer. The second orders half a beer. The third, a quarter of a beer. The bartender says “You’re all idiots”, and pours two beers.
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Vlad Matei
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Fie \( F \) o primitiva al lui \( f \).Avem ca \( \displaystyle \int_{0}^{1} f(x) dx=F(1)-F(0) \) deoarece \( f \) continua. Fie \( \displaystyle g(x)=\frac{F(x)-F(0)}{e^x} \).Avem ca \( \displaystyle g(1)=g(0)=0 \) de unde aplicand Rolle rezulta ca exista \( c\in (0;1) \) astfel incat \( g'(c)=0 \) de unde se obtine \( \displaystyle f(c)=\int_{o}^{c} f(x) dx \)
Last edited by Vlad Matei on Thu Sep 27, 2007 6:25 pm, edited 1 time in total.