Answer
a) $11$ and $12$
b) $11.5$ and $11.5$
Work Step by Step
a) We needn’t consider pairs where the first number is larger than the second, since we can just interchange the numbers in such cases. The answer appears to be 11 and 12, but we have considered only integers in the table.
b) Let's note the two numbers by $x$ and $y$.
Then $x+y$ = $23$ so $y$ = $23-x$
$P(x)$ = $xy$ = $x(23-x)$ = $23x-x^{2}$
$P'(x)$ = $23-2x$
$P'(x)$ = $0$ then $x$ = $\frac{23}{2}$ = $11.5$
So the maximum value of $P$ is $P(11.5)$ = $11.5^{2}$ = $132.25$
since $P''(x)$ = $-2$ $\lt$ $0$ for all $x$ so $P$ is everywhere concave downward and local maximum at $x$ = $11.5$ is an absolute maximum.
