Divide 64 into two parts such that the sum of the cubes of two parts is minimum.

Let the two positive numbers be $a$ and $b$

Given \[a\text{ }+\text{ }b\text{ }=\text{ }64\text{ }\ldots \text{ }\left( 1 \right)\]

We have, \[{{a}^{3}}~+\text{ }{{b}^{3}}\] is minima

Assume, \[S\text{ }=\text{ }{{a}^{3}}~+\text{ }{{b}^{3}}\]

(From equation 1)

\[S\text{ }=\text{ }{{a}^{3}}~+\text{ }{{\left( 64\text{ }-\text{ }a \right)}^{3}}\]