The sum of two numbers is fixed. Their product is maximum when:

Let the two numbers be $x$ and $y$. Let their fixed sum be $S$.
$$x+y=S⟹y=S-x$$
The product function $P(x)$ is:
$$P(x)=x\cdot y=x(S-x)=Sx-x^2$$
To find the maximum, find the first derivative and set it to zero:
$$P^{\prime }(x)=S-2x=0⟹x=\frac{S}{2}$$
Check the second derivative:
$$P^{\prime \prime }(x)=-2<0$$
Since $P^{\prime \prime }(x)$ is negative, the product is maximum at $x=S/2$.Then $y=S-S/2=S/2$.Thus, each number is half of the sum.