# Question e05e2

Aug 11, 2017

$13 + 13 = 26$

$13 \times 13 = 169$

#### Explanation:

Lets arrange pairs of numbers which add to $26$ in order and find their product.

$0 + 26 : \text{ } \rightarrow 0 \times 26 = 0$
$1 + 25 : \text{ } \rightarrow 1 \times 25 = 25$
$2 + 24 : \text{ } \rightarrow 2 \times 24 = 48$
$3 + 23 : \text{ } \rightarrow 3 \times 23 = 69$

$7 + 19 : \text{ } \rightarrow 7 \times 19 = 133$
$10 + 16 \text{ } \rightarrow 10 \times 16 = 160$

We notice that as the numbers get closer, the difference between them gets less, but their product increases.

The smallest difference will be when the two numbers are equal and their product will therefore be the biggest.

$12 + 14 : \text{ } \rightarrow 12 \times 14 = 168$
$13 + 13 : \text{ } \rightarrow 13 \times 13 = 169$

Aug 11, 2017

The numbers are $13$ and $13$

#### Explanation:

Let one number be $= x$

Then, the other number is $= 26 - x$

Let the product of the numbers be $= y$

So,

$y = x \left(26 - x\right) = 26 x - {x}^{2}$

Differentiating with respect to $x$

$y ' = \frac{\mathrm{dy}}{\mathrm{dx}} = 26 - 2 x$

The critical value is when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\implies$, $26 - 2 x = 0$

$\implies$, $x = 13$

Let's make a variation chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a}$$13$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$y '$$\textcolor{w h i t e}{a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a}$↗$\textcolor{w h i t e}{a a a a}$$0$$\textcolor{w h i t e}{a a a a}$↘#

There is a maximum at $x = 13$

We confirm this by calculating the second derivative

$y ' ' = \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = - 2$

As $y ' ' < 0$, we conclude that when $x = 13$, there is a maximum value