How should we choose two numbers, each greater than or equal to –2, whose sum is ½ so that the sum of the first and the cube of the second is minimum?

As per the given question,

\[1\text{ }+\text{ }3{{\left( {\scriptscriptstyle 1\!/\!{ }_2}\text{ }-\text{ }a \right)}^{2}}~\left( -1 \right)\text{ }=\text{ }0\]

\[1\text{ }-\text{ }3{{\left( {\scriptscriptstyle 1\!/\!{ }_2}\text{ }-\text{ }a \right)}^{2}}~=\text{ }0\]