Reality.......

Algebra Level 2

If a + b + c a+b+c = 1 1 and a , b , c > 0 a,b,c>0 Then what is the least value of 1 / a + 1 / b + 1 / c 1/a+1/b+1/c


The answer is 9.

Kartik Umate by Kartik Umate , 24, India

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2 solutions

Qi Huan Tan
Jan 29, 2015

By Cauchy-Schwarz inequality, ( a + b + c ) ( 1 a + 1 b + 1 c ) ≥ ( 1 + 1 + 1 ) 2 = 9 (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq (1+1+1)^2=9 . Equality occurs when a = b = c = 1 3 a=b=c=\frac{1}{3} .

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Kalpok Guha
Dec 7, 2014

Arithmetic Mean of a,b,c is (a+b+c)/3 and Harmonic mean of a,b,c is 3/(1/a+1/b+1/c).By AM-HM inequality We get (a+b+c)/3 ≥ 3/(1/a+1/b+1/c).Using a+b+c=1 and rearranging the expression we get (1/a+1/b+1/c) ≥ 9 So the least value of 1/a+1/b+1/c is 9.

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