If a, b, c are positive numbers and a + b + c = 1 , then the maximum value of (1 - a)(1 - b)(1 - c) is:

Step-by-step solution

We know that a+b+c=1 . Then 1-a = b+c , 1-b = a+c , and 1-c = a+b . We want to maximize (b+c)(a+c)(a+b) . By AM-GM inequality: b+c 2 bc a+c 2 ac a+b 2 ab Multiplying them gives (b+c)(a+c)(a+b) 8abc . Alternatively, using AM-GM on (1-a), (1-b), (1-c) : (1-a)+(1-b)+(1-c) 3 [3] (1-a)(1-b)(1-c) 3-(a+b+c) 3 [3] (1-a)(1-b)(1-c) 2 3 [3] (1-a)(1-b)(1-c) Taking the cube on both sides: (1-a)(1-b)(1-c) 8 27