The sum of two numbers is radical 3. Find the largest possible value for their product
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To find the largest possible value for the product of two numbers whose sum is √3, we need to use the concept of maximizing a product.
Let the two numbers be x and y. According to the given condition, we have the equation:
x + y = √3
To maximize the product xy, we can use the AM-GM inequality, also known as the Arithmetic Mean-Geometric Mean inequality. According to this inequality, the geometric mean of two positive numbers is always less than or equal to their arithmetic mean. Mathematically, it can be written as:
√(xy) ≤ (x + y)/2
In this case, since the sum of x and y is √3, we can rewrite the inequality as:
√(xy) ≤ √3/2
Now, we want to find the largest possible value for the product xy. To do that, we square both sides of the inequality:
xy ≤ (√3/2)²
xy ≤ 3/4
Therefore, the largest possible value for the product of two numbers whose sum is √3 is 3/4.