Answer
(a) It seems that the product will reach a maximum when the first number is $11.5$ and the second number is $11.5$
(b) The two numbers are $11.5$ and $11.5$
Work Step by Step
(a) Based on the numbers in the table, it seems that the product will reach a maximum when the first number is $11.5$ and the second number is $11.5$
(b) Let $x$ and $y$ be the two numbers.
$x+y = 23$
$x = 23-y$
We can write an expression for the product:
$P = xy = (23-y)(y) = 23y - y^2$
We can find the point where $P'(y) = 0$:
$P'(y) = 23-2y = 0$
$2y = 23$
$y = 11.5$
Note that $P''(y) = -2 \lt 0$
Since this graph is concave downward, $y=11.5$ must be the point where the product is a maximum.
We can find $x$:
$x = 23-(11.5) = 11.5$
The two numbers are $11.5$ and $11.5$
This is the same as our answer in part (a).
