Mini-Minimum Value

Algebra Level 2

Let a , b a,b and c c be positive real numbers. Find the minimum value of a b + b c + c a \dfrac a b + \dfrac b c + \dfrac c a .


The answer is 3.

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Jim Chale by Jim chale , 20, Greece

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1 solution

Samrit Pramanik
Jul 8, 2018

By AM-GM inequality we have

a b + b c + c a 3 ≥ ( a b b c c a ) 1 3 = 1 \displaystyle \frac{\displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{3} \geq \displaystyle \left(\frac{a}{b}\frac{b}{c}\frac{c}{a}\right)^{\frac{1}{3}}=1

Hence, a b + b c + c a ≥ 3 \displaystyle \frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \boxed{3}

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