if  x and y are two positive real numbers whose product is 100,  x*y = 100, 

    then the the minimum value of x+y is achieved at x = y = 10.

If x*y = 100,  then  y = ,  and the sum  x+y  is  x + .


This function of "x",  f(x) = x +   achieves the minimum when its derivative is equal to zero


    f'(x) = 1 -  = 0.


Then    = 100;  hence  x =  = 10.


Thus we proved that if  x*y = 100,  then  x+y has minimum at  x = y = 10.