12. The price of a diamond is proportional to the square of its mass. If a diamond is broken into two parts, in which case will the total price of the two parts be the lowest?
12. Solution. Let the price of a diamond be calculated by the formula \(y=a m^{2}\), where \(m\) is its mass. Let the mass of the first piece be \(\frac{m}{2}+x\). Then the mass of the second piece will be \(m-\frac{m}{2}-x=\frac{m}{2}-x\). The total cost of the two pieces will be:

\[
a\left(\frac{m}{2}+x\right)^{2}+a\left(\frac{m}{2}-x\right)^{2}=a\left(\frac{m^{2}}{4}+x^{2}\right)
\]

Obviously, this sum will be the smallest if \(x=0\), i.e., if both pieces have the same mass.