If the product of two numbers is 25, is it possible to determine a minimum value for the sum of the numbers?

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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

x*y=25

=> y = 25/x

x+y = x + 25/x

We can use the arithmetic-geometric mean formula, which states that

(a+b)/2>=(ab)^(1/2)

=> a+b>=2*(ab)^(1/2)

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

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