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 can use the concept of quadratic optimization.
Let's assume the two numbers are x and y. We are given that x + y = √3. We want to find the maximum possible value of xy.
To solve this problem, we will use the AM-GM inequality. According to the AM-GM inequality, the arithmetic mean of a set of positive numbers is always greater than or equal to their geometric mean.
In our case, the arithmetic mean of x and y is equal to (x + y)/2 = (√3)/2. The geometric mean of x and y is equal to √(xy).
So, we have (√3)/2 ≥ √(xy).
Now, let's square both sides of the inequality to get rid of the square root:
((√3)/2)^2 ≥ (√(xy))^2
(3/4) ≥ xy
Therefore, the largest possible value for xy is 3/4. To achieve this maximum value, x and y must be equal.
In conclusion, the largest possible value for the product of two numbers whose sum is √3 is 3/4, and this maximum is attained when the two numbers are equal.