MQ73 --- Inequality

By A.M. ≥ G.M. ,
3⁻ᵃ + 9⁻ᵇ + 27⁻ᶜ = 3⁻ᵃ + 3⁻²ᵇ + 3⁻³ᶜ
≥ 3∛3⁻ᵃ ⁻²ᵇ ⁻³ᶜ = 3 * 3⁽⁻ᵃ ⁻²ᵇ ⁻³ᶜ⁾ˡ ³
By Cauchy-Schwarz's inequality ,
(a + 2b + 3c)²
= (a + √2 √2b + √3 √3c)² ≤ (1 + 2 + 3)(a² + 2b² + 3c²)
= 6 * 3/2 = 9.

⇒ a + 2b + 3c ≤ 3
i.e. - a - 2b - 3c ≥ -3
∴ 3⁻ᵃ + 9⁻ᵇ + 27⁻ᶜ ≥ 3 * 3⁽⁻ᵃ ⁻²ᵇ ⁻³ᶜ⁾ˡ ³ ≥ 3 * 3⁻³ˡ ³ = 1.

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