Prove that 1/a + 1/b + 1/c >= 9 for any a,b,c >0 such that a + b + c =1no.

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sciencesolve eNotes educator| Certified Educator

You need to use the formulas of harmonic mean and arithmetic mean of three positive numbers such that:

Arithmetic mean = `(a+b+c)/3`

Harmonic mean = `3/(1/a + 1/b + 1/c)`

You need to remember that arithmetic mean is larger than harmonic mean such that:

`(a+b+c)/3 gt= 3/(1/a + 1/b + 1/c) =gt 3/(a+b+c) =lt (1/a + 1/b + 1/c)/3`

Substituting 1 for a+b+c yields:

`3=lt (1/a + 1/b + 1/c)/3`

You need to multiply by 3 both sides such that:

`9 =lt (1/a + 1/b + 1/c)`

This last line proves the inequality `(1/a + 1/b + 1/c) gt= 9.`