Question

Find three positive numbers \(x, y,\) and \(z\) that satisfy the given conditions. The sum is 30 and the sum of the squares is a minimum.

Short answer

The numbers that minimize the sum of squares under the given conditions are \(x = y = z = 10\), and the minimum sum of squares is 300.

Step-by-step solution

Formulate the Equations

The problem gives the following information:\nThe sum of the three numbers is 30: \(x + y + z = 30\).\nThe goal is to minimize the sum of their squares, say \(S = x^2 + y^2 + z^2\), subject to the above constraint. Here S is our objective function.

Use the Constraint to Simplify the Objective Function

By the constraint, we can write \(z = 30 - x - y\).\nSubstitute this into the objective function \(S\), we obtain \n\(S = x^2 + y^2 + (30 - x - y)^2\).

Minimize the Objective Function

To find the minimum of the function \(S\), we differentiate \(S\) with respect to \(x\) and \(y\) separately, and set each to zero to find the stationary points. As a result, we obtain two equations, solve them, we get the solution \(x = y = z = 10\).

Verify the Solution

Verify that \(S\) is indeed minimized at this point by showing that the second order derivatives are positive, which means that \(S\) is convex and thus has a minimum at the point \(x = y = z = 10\). Hence the minimum sum of squares is \(S = 3 * 10^2 = 300\).