Question 3
(a) Two nonnegative numbers $x,y$ have fixed sum $a$. Determine $x,y$ for which $x^m+y^m$ is smallest, where $m$ is a given natural number. Given $n$ nonnegative numbers with fixed sum, prove that for $x_1^m+x_2^m+\cdots+x_n^m$ to be smallest one must have $x_1=x_2=\cdots=x_n$.
(b) One cubic meter of a chemical powder is to be stored in eight cubic wooden boxes. Every flat wooden panel has the same fixed, negligible thickness. The price of each square panel of that thickness is proportional to the square of its area. How should the edges $x_1,x_2,\ldots,x_8$ be chosen to minimize the cost of wood?
(b) One cubic meter of a chemical powder is to be stored in eight cubic wooden boxes. Every flat wooden panel has the same fixed, negligible thickness. The price of each square panel of that thickness is proportional to the square of its area. How should the edges $x_1,x_2,\ldots,x_8$ be chosen to minimize the cost of wood?