Question

The sum of two positive numbers is given. If the sum of their cubes is minimum, then

• A. They are equal
• B. One is twice the other
• C. They are unequal
• D. One is thrice the other

Answer 1

The correct option is A They are equal

Let the sum of two positive numbers be x and the two numbers be a and b.

$x=a+b$

$x^3=a^3+b^3=(a+b)\left[\right(a+b{)}^2-3ab]$

As we can see that the sum of cubes is dependent on the product of the two numbers, since their sum is given to be $x$.

Also, their product term has been subtracted and so to obtain the minimum sum of their cubes, we need to maximize their product.

Using $A.M.\ge G.M.$, we have $a+b\ge 2\sqrt{ab}$

The maximum value of product is obtained when the numbers are equal.