Find three positive integers x, y, and z that satisfy the given conditions. the product is 64, and the sum is a minimum.

Minimize subject to . Using Lagrange multipliers, we can take the Lagrangian



which has partial derivatives (set to 0)



From the first three equations, pick any two and subtract them from one another. You'll arrive at the following symmetric relations:



We assume , and we know that so that we can omit the possibilities of , , and . This leaves us with , and so

.