Answer
The pair is $8$ and $8$.
The maximum product is $64$.
Work Step by Step
The given sum of pairs of numbers is $16$.
Let's note by $x$ one of the numbers of the pair whose product is largest.
Because the sum of the two numbers is $16$, the other number is $16-x$.
The product is
$\Rightarrow P(x)=x(16-x)$
Use the distributive propery.
$\Rightarrow P(x)=16x-x^2$.
Rearrange.
$\Rightarrow P(x)=-x^2+16x$.
The standard form of a quadratic equation is $f(x)=ax^2+bx+c$.
Compare both equations and identify the constants $a=-1$, $b=16$, $c=0$.
The value of $a$ is less than zero, the function has the maximum value.
The value of $x$ for which the maximum is reached is $-\frac{b}{2a}$.
Substitute all values.
$\Rightarrow x=-\frac{16}{2(-1)}$.
Simplify.
$\Rightarrow x=8$.
The other number is $16-x=16-8=8$.
Hence, the pair is $8$ and $8$.
The maximum product is $8\cdot 8=64$.
