Exercise 6.3 · Q13: Sum 24, largest product: the numbers $12$ and $12$

Basics you need

Solution

1. Choose the variable and its allowed range

   Let one number be $x$⁠. Since the sum is 24, the other number is $24 - x$⁠. The question puts no other restriction on the numbers, so

   $$x \in \mathbf{R}\quad(\text{any real number}).$$
   Every pair of numbers with sum 24 has the form $x$ and $24 - x$ for exactly one $x$ (take $x$ to be the first number). So maximising over all real $x$ is the same as maximising over all such pairs.
   From $x + y = 24$⁠, subtract $x$⁠: $y = 24 - x$⁠. Basics: building the function

2. The product as a function of $x$

   $$P(x) = x(24 - x) = 24x - x^2.$$
   Product means multiply the two numbers. Expanding makes it easy to differentiate term by term.
   $x(24 - x) = 24x - x^2$ by the distributive law. Basics: algebra

3. Critical point

   $$P'(x) = 24 - 2x, \qquad P''(x) = -2.$$ $$P'(x) = 0 \iff 2x = 24 \iff x = 12.$$
   $P$ is a polynomial, differentiable everywhere, so its only critical point is where $P'(x) = 0$⁠, namely $x = 12$⁠.
   Critical point: $f'(c) = 0$ or $f$ not differentiable (Note after Theorem 2, Section 6.4). Lesson: maxima and minima

4. Second derivative test

   $P''(12) = -2 < 0$⁠, so by the second derivative test (Theorem 4) $x = 12$ is a point of local maxima of $P$⁠.

   $P'(12) = 0$ and $P''(12) < 0$⁠: the graph is flat at $x = 12$ and bends downward there, so it is a peak.
   Theorem 4(i), Section 6.4. Lesson: derivative tests

5. It is the largest product of all

   The sign of $P'(x) = 2(12 - x)$ is the sign of $12 - x$⁠:

   • for $x < 12$⁠: $P'(x) > 0$⁠, so $P$ is increasing on $(-\infty, 12]$⁠;
   • for $x > 12$⁠: $P'(x) < 0$⁠, so $P$ is decreasing on $[12, \infty)$⁠.

   Hence $P(x) < P(12)$ for every real $x \ne 12$⁠. The same fact can be seen without calculus by completing the square:

   $$P(x) = 144 - (x - 12)^2 \le 144,$$

   with equality only at $x = 12$⁠.

   The second derivative test only speaks about points near 12. The question asks for the largest product overall, so we need the global argument: $P$ rises all the way up to 12 and falls all the way after it.
   $144 - (x - 12)^2$ $= 144 - x^2 + 24x - 144$ $= 24x - x^2$⁠. Basics: sign of a product

6. Answer the question

   $x = 12$⁠, and the other number is $24 - 12 = 12$⁠. The largest product is $P(12) = 12 \times 12 = 144$⁠.

   The question asks for the two numbers, so state both, with the product as a check.
   Answer in the words of the question. Basics: building the function

Every fact used, and where it comes from