# Help find answer 18?

##### 1 Answer
Jan 22, 2018

$5$

#### Explanation:

let $x =$one of the positive integers
that means $10 - x$ is the other positive integer since they add to 10.

the product of those numbers is $x \left(10 - x\right) = - {x}^{2} + 10 x$
you want to maximize the value $- {x}^{2} + 10 x$

this graph is a parabola opening down, as you can see from the -1 coefficient on the ${x}^{2}$ term.

the maximum value of $- {x}^{2} + 10 x$ occurs at the vertex, or where the derivative = 0.

$\frac{d}{\mathrm{dx}} \left(- {x}^{2} + 10 x\right) = - 2 x + 10$
$- 2 x + 10 = 0$ when $x = 5$

this means the vertex of the graph is $\left(5 , 25\right)$, meaning the two positive integers are 5 and 10-5, or simply 5 and 5.

the answer is $5$.