Example 22: Sum 15, least sum of squares: both numbers $\tfrac{15}{2}$

Basics you need

Solution

1. One variable and its allowed range

   Let the first number be $x$⁠. The sum is 15, so the second number is $15 - x$⁠. Both must be positive:

   $$x > 0 \quad\text{and}\quad 15 - x > 0 \iff 0 < x < 15.$$
   The condition "sum is 15" removes one unknown. The word "positive" fixes the range of $x$⁠; we need the range later to be sure that our answer is the least value over all allowed choices, not just near one point.
   $15 - x > 0$ rearranges to $x < 15$⁠. Basics: interval notation

2. Write the sum of squares as a function of $x$

   $$\begin{aligned} S(x) &= x^2 + (15 - x)^2 \\ &= x^2 + 225 - 30x + x^2 \\ &= 2x^2 - 30x + 225, \quad 0 < x < 15. \end{aligned}$$
   This is the quantity the question wants as small as possible. Expanding makes it easy to differentiate.
   $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 15$⁠, $b = x$⁠. Basics: building the function to optimise

3. Differentiate and find the critical point

   $$S'(x) = 4x - 30, \qquad S''(x) = 4.$$ $$S'(x) = 0 \iff x = \frac{30}{4} = \frac{15}{2}.$$

   $\frac{15}{2} = 7.5$ lies inside $(0, 15)$⁠, so it is an allowed value. $S$ is a polynomial, differentiable everywhere, so this is the only critical point.

   By Theorem 2 the smallest value at an interior point can only occur where $S' = 0$ (or $S$ is not differentiable, which never happens here). Checking that $\frac{15}{2}$ is inside the range matters: a critical point outside the range would be useless.
   Power rule: $\frac{d}{dx}(2x^2) = 4x$⁠, $\frac{d}{dx}(-30x) = -30$⁠, $\frac{d}{dx}225 = 0$⁠.

4. Second derivative test: a local minimum

   $$S''\!\left(\tfrac{15}{2}\right) = 4 > 0.$$

   By the second derivative test, $x = \frac{15}{2}$ is a point of local minima of $S$⁠.

   A positive second derivative means the slope $S'$ is increasing: negative before $\frac{15}{2}$⁠, zero at it, positive after. That is the shape of a valley.
   Theorem 4(ii): $S'(c) = 0$ and $S''(c) > 0$ give a point of local minima. Lesson: first and second derivative tests

5. It is the least value on the whole range

   "Local" only compares nearby points. To compare with every allowed $x$⁠, look at the sign of $S'(x) = 4x - 30 = 4\left(x - \frac{15}{2}\right)$⁠:

   • for $0 < x < \frac{15}{2}$⁠, $S'(x) < 0$⁠, so $S$ is decreasing there;
   • for $\frac{15}{2} < x < 15$⁠, $S'(x) > 0$⁠, so $S$ is increasing there.

   So $S$ goes down all the way to $x = \frac{15}{2}$ and up all the way after it: $S(x) > S\!\left(\frac{15}{2}\right)$ for every other $x$ in $(0, 15)$⁠. The same follows from completing the square:

   $$S(x) = 2\left(x - \tfrac{15}{2}\right)^2 + \tfrac{225}{2} \ \ge\ \tfrac{225}{2}.$$
   The book stops at the second derivative test, which strictly speaking gives only a local minimum. With one critical point and $S'$ negative before it and positive after it, the local minimum is also the least value overall. The ends 0 and 15 are excluded, and near them $S$ is close to 225, far above $\frac{225}{2}$⁠, so nothing better hides there.
   Theorem 1, Section 6.3: $f' < 0$ on an interval gives decreasing, $f' > 0$ gives increasing. Lesson: increasing and decreasing functions

6. Answer the question asked

   The first number is $x = \frac{15}{2}$ and the second is $15 - \frac{15}{2} = \frac{15}{2}$⁠. The least sum of squares is

   $$S\!\left(\tfrac{15}{2}\right) = \tfrac{225}{4} + \tfrac{225}{4} = \tfrac{225}{2} = 112.5.$$
   The question asks for the two numbers, so we name both. The least value itself is a useful extra.

7. The general rule (the book's Remark)

   Replace 15 by any positive number $k$⁠. Then $S(x) = x^2 + (k-x)^2$⁠, $S'(x) = 4x - 2k$⁠, $S''(x) = 4 > 0$⁠, and $S'(x) = 0$ gives $x = \frac{k}{2}$⁠. So the two positive numbers with sum $k$ and least sum of squares are $\frac{k}{2}$ and $\frac{k}{2}$⁠.

   The book's Remark says "Proceeding as in Example 34". Example 34 in this reprint is a different problem (it shows that $\tan^{-1}(\sin x + \cos x)$ is increasing), so the reference is left over from an older numbering; the example meant is this one (Example 22). The working above is exactly that "proceeding".

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