If x+y =10 What is the minimum value of 1/xy

To find the minimum value of 1/xy given that x + y = 10, we can use the AM-GM inequality (Arithmetic Mean-Geometric Mean Inequality).

Step 1: The AM-GM inequality states that the arithmetic mean of a set of non-negative numbers is always greater than or equal to the geometric mean. In this case, we have two numbers, x and y, and their sum is 10. So, the arithmetic mean is (x + y)/2 = 10/2 = 5.

Step 2: The geometric mean is the square root of the product of the numbers, which in this case is √(xy). According to the AM-GM inequality, 5 >= √(xy).

Step 3: Squaring both sides of the inequality to get rid of the square root gives us 25 >= xy.

Step 4: Now, if we take the reciprocal of both sides, we have to reverse the inequality sign (because we are dealing with positive numbers), so we get 1/25 <= 1/xy.

So, the minimum value of 1/xy is 1/25.