QPaperGen

Let one of the numbers be \( x \).
Then, the other number is \( 24 - x \).

Let the product of the two numbers be:
\[
P(x) = x(24 - x) = 24x - x^2
\]

Differentiate \( P(x) \):
\[
P'(x) = 24 - 2x, \quad P''(x) = -2
\]

Set the first derivative to zero for critical points:
\[
P'(x) = 0 \Rightarrow 24 - 2x = 0 \Rightarrow x = 12
\]

Check the second derivative:
\[
P''(12) = -2 < 0
\]

Since \( P''(x) < 0 \), the function attains a **maximum** at \( x = 12 \).

Therefore, the two numbers are:
\[
x = 12, \quad 24 - x = 12
\]


The two numbers are 12 and 12. The product is maximum when both are equal.