Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.

Let the two positive numbers are  and 

  

     ……….(i)

Let 

  [From eq. (i)]

 

= 

  and 

Now  = 0

 

 

At    is positive.

   is a point of local minima and  is minimum when .

  

Therefore, the required numbers are 8 and 8.