2021/092) The sum of two numbers is 15. What is the minimum sum of the resultant cubes of the two numbers?

Let the numbers be x and y

We have x+ y = 15

$(x+y)^3 = x^3 + y^3 + 3xy(x+y)$

or $15^3 = x^3 + y^3 + 3xy * 15$

or $x^3 + y^3 = 15^3 - 45xy$

this is mininum when xy is maximum

$x + y = 15$

we have $4xy = (x+y)^2 - (x-y)^2 = 15 - (-x-y)^2$

so xy is maximumum when x = y

or $x^3 + y^3$ is minumum when $x = y = 7.5$ and value is $2 * 7.5^3 = 843.75$