Question

Find three positive numbers whose sum is 12 and the sum of whose squares is as small as possible.

Short answer

The numbers are\(4,4,4\).

Step-by-step solution

Second derivative test

Suppose the second partial derivatives of\(f\)are continuous on a disk with center\({\bf{(a,b)}}\), and suppose that\({f_x}(a,b) = 0\)and\({f_y}(a,b) = 0\). Let

\(D = D(a,b) = {f_{xx}}(a,b){f_{yy}}(a,b) - {\left( {{f_{xy}}(a,b)} \right)^2}\)

(a) If\(D > 0\)and\({f_{xx}}(a,b) > 0\), then\(f(a,b)\)is a local minimum.

(b) If\(D > 0\)and\({f_{xx}}(a,b) < 0\), then\(f(a,b)\)is a local maximum.

(c) If\(D < 0\), then\(f(a,b)\)is not a local maximum or minimum.

Form equation

Let the three numbers be\(a,b\), and\(c\).

According to question\(f(a,b,c) = {a^2} + {b^2} + {c^2}\)& the constraints are:

\(\begin{array}{l}a \ge 0\\b \ge 0\\c \ge 0\\a + b + c = 12\end{array}\)

We have

\(\begin{array}{l}c = 12 - a - b\\f = {a^2} + {b^2} + {(12 - a - b)^2}\end{array}\)

Simplifying & collecting like terms.

\(\begin{array}{c}f = {a^2} + {b^2} + \left( {144 - 12a - 12b - 12a + {a^2} + ab - 12b + ab + {b^2}} \right)\\ = 2{a^2} + 2{b^2} + 2ab - 24a - 24b + 144\end{array}\)

Compute partial derivative

The partial derivatives.

\(\begin{array}{l}{f_a} = 4a + 2b - 24\\{f_b} = 4b + 2a - 24\end{array}\)

Find critical point

Put\({f_a} = 0,{f_b} = 0\)

\(\begin{array}{r}0 = 4a + 2b - 24\\ = 2a + b - 12\\b = 12 - 2a\end{array}\)

Continue with the partial with respect to\(b\)and substituting the result above.

\(0 = 4b + 2a - 24\)

\(\begin{array}{c}0 = 2b + a - 12\\12 = 2(12 - 2a) + a\\12 = 24 - 3a\\a = 4\end{array}\)

And\(b = 4\).

Substitute values we get\(c = 4\).

Compute partial derivative

The second derivative test requires all the second partials.

\(\begin{array}{l}{f_{aa}} = 4\\{f_{bb}} = 4\\{f_{ab}} = 2\end{array}\)

Compute the quantity\(D\).

\(\begin{array}{c}D = {f_{aa}}{f_{bb}} - {\left( {{f_{ab}}} \right)^2}\\ = 12\end{array}\)

Since this is positive this point is an extreme value. Since\({f_{aa}} > 0\)the point is a minimizer.

The numbers\(4,4,4\)are positive, add to twelve, and minimize the sum of the squares.