If the product of two numbers is 25, is it possible to determine a minimum value for the sum of the numbers?
The product of two numbers is given to be 25. If one of the numbers is x, the other is `25/x` . The sum of the two numbers is `S = x + 25/x`
Now, a minimum value of x can be found at S' = 0 and if S'' is positive.
`S' = 1 - 25/x^2`
`S'' = 25/x^3`
Solving S' = 0
=> `25/x^2 = 1`
=> x = +5 and x = -5
For x = -5, S'' is negative and for x = 5, S'' is positive.
The minimum value for the sum of the numbers is at x = 5 and is equal to 10.
Let two number be x and y
=> y = 25/x
x+y = x + 25/x
We can use the arithmetic-geometric mean formula, which states that
plugging in x and 25/x for a and b,
x+25/x >=2*(x*25/x)^(1/2) = 2 * 25^(1/2) = 10
Therefore, the minimum value of the sum of two number x and y is 10.
And the minimum is achieved when x = y
=> x = 25/x
=> x^2 = 25
=> x = 5
x+y = 10 when x and y are both 5