Solution to Show that for any positive number a and b, (a + b)/2 ≥ √ab - Sikademy

Nov. 28, 2020

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Archangel Macsika

Show that for any positive number a and b, (a + b)/2 ≥ √ab

Solution

Proof.

(a + b)/2 ≥ √ab

(a + b)/2 ≥ √ab ⟺ ((a + b)/2)2 ≥ ab

⟺ (a + b)2 = 4ab

⟺ a2 + b2 + 2ab ≥ 4ab

⟺ a2 − 2ab + b2 ≥ 0

⟺ (a − b)2 ≥ 0

Since square on any number is greater than zero, so the (a − b)2 ≥ 0 and so we have the inequality.

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