Final answer:
The largest product of positive numbers x, y, and z given the condition that x + y + z^2 = 16 is 32, achieved when x = y = 4 and z = 2.
Step-by-step explanation:
The student is asking to find the largest product of positive numbers x, y, and z under the condition that x + y + z^2 = 16. To solve this, we need to use the AM-GM inequality, which states that for non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. The equality holds when all numbers are equal. Therefore, x and y need to be as close to each other as possible, and as close to z^2as possible, while their sum is 16. As the product is maximized when the numbers are equal, given the constraint, the largest z^2 (and thus z) would be when x and y are equal and the sum of x, y, and z^2 is 16. Solving for this, we will find x = y = 4 and z = 2, leading to the largest product of 4 * 4 * 2 = 32.