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Find the minimum of S = x + y, the sum of two positive numbers whose product is 800, that is xy = 800

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Final answer:

The minimum value of S = x + y, given xy = 800, is found using the AM-GM Inequality, resulting in the minimum sum S being 40√2 when x and y are both equal to 20√2.

Step-by-step explanation:

To find the minimum value of S = x + y, given that the product of two positive numbers is xy = 800, we can use the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality). The AM-GM Inequality states that for non-negative numbers x and y, the arithmetic mean is always greater than or equal to the geometric mean.

In this case, we can apply the inequality as follows:

  • Arithmetic Mean (AM) of x and y = (x+y)/2
  • Geometric Mean (GM) of x and y = √(xy)
  • AM ≥ GM
  • (x+y)/2 ≥ √(800)
  • S/2 ≥ √(800)
  • S ≥ 2√(800)
  • S ≥ 2 * 20 √2
  • S ≥ 40√2

Therefore, the minimum value of S is 40√2 when x and y are equal, which means x = y = √(800) = 20√2.

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