Minim de maxim de a,b,c cu conditie a+b+c=abc

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alex2008
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Minim de maxim de a,b,c cu conditie a+b+c=abc

Post by alex2008 »

Fie \( a,b,c>0 \) astfel incat \( a+b+c=abc \). Sa se gaseasca valoarea minima a lui \( \max \{a,b,c\} \).
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DrAGos Calinescu
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Post by DrAGos Calinescu »

Evident din AM-GM \( abc\ge\sqrt{27} \)
Deci \( \max\{a,b,c\}\ge\sqrt{3} \)
Dar pentru \( a=b=c=\sqrt{3} \) egalitatea este adevarata, deci minimul cautat este \( \sqrt{3} \).
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