Prove-that-abc-1-a-1-b-1-c-1-for-0-a-b-c-1-

Question Number 63247 by Joel122 last updated on 01/Jul/19

Prove that

\sqrt{a b c} + \sqrt{(1 - a) (1 - b) (1 - c)} ⩽ 1

for 0 ⩽ a, b, c ⩽ 1

Commented by Tony Lin last updated on 01/Jul/19

∵ a b c + (1 - a) (1 - b) (1 - c) ⩾

{[\sqrt{a b c} + \sqrt{(1 - a) (1 - b) (1 - c)}]}^{2}

(u s i n g {C a u c h y}^{'} s i n e q u a l i t y)

(1 - a) (1 - b) (1 - c) + a b c

= 1 - c - a - b + a c + b c + a b ⩽ 1

∴ \sqrt{a b c} + \sqrt{(1 - a) (1 - b) (1 - c)} ⩽ 1