Final answer:
To maximize the product x * y * z under the constraint x + y + z = 1 with x, y, z > 0, the maximum value is attained when x = y = z = 1/3, resulting in the maximum product of 1/27.
Step-by-step explanation:
The student is asking how to maximize the product of three positive numbers (x, y, z) given the constraint that their sum is equal to 1. This is a problem in optimization and can be approached using methods from calculus or even by employing the arithmetic mean-geometric mean inequality which states that the geometric mean of a set of positive numbers is less than or equal to the arithmetic mean of those numbers. Since x, y, and z must be positive and adhere to the constraint x + y + z = 1, we can apply this inequality.
By the inequality, we have:
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- (x * y * z)^(1/3) ≤ (x + y + z) / 3
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- (x * y * z)^(1/3) ≤ 1/3
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- x * y * z ≤ (1/3)^3
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- x * y * z ≤ 1/27
This inequality becomes an equality when x = y = z, which happens at x = y = z = 1/3. Therefore, the maximum product occurs when x, y, and z are all equal to 1/3, and the maximum value of the product x * y * z is 1/27.