Question

If a ≥ 0 and b ≥ 0, prove that (a+b)/2 ≥ sqrt(ab) with equality if and only if a = b.

Answer 1

This is one of my favorite results from elementary mathematics

I will write > for less than or equals, just because I don't have than on my keyboard.

(a+b)/2 > sqrt(ab)
if and only if (iff)
a + b > 2 sqrt(ab)
iff
(a+b)^2 > 4 ab
iff
a^2 + 2ab + b^2 > 4ab
iff
a^2 - 2ab + b^2 > 0
iff
(a-b)^2 > 0

Notice now that the equality holds
(a-b)^2 = 0
if and only if
a - b = 0

Got it?

Answer 2

Awesome. I love this prove. Thanks!