What is the minimum product of $a$ and $b$? $a$ and $b$ are positive numbers
$$a + b = 13$$
According to this, what is the minimum product of $a$ and $b$?
I'm trying to find the easiest way to compute it. 
Regards
 A: $b=13-a$
product is $p(a)=a(13-a)$
if $a$ varies from $0$ to $13$ then product $p$ varies from $0$ to $\dfrac{169}{4}$ when $a=\dfrac{13}{2}$
If you want $a,b$ positive the domain of $a$ is open that is $0<a\le 13$ therefore there exist no minimum for the product but just the so called infimum 
This means that the product goes down near to zero but is never zero because $a>0$
Minimum does not exist
Hope this helps
$$...$$

A: Since $a+b=13$, then $b=13-a$ and $ab=a(13-a)$ is a quadratic trinomial with a maximum at the vertex of parabola, i.e. for $a=6.5$. The value is $6.5^2=42.25$. The minimum does not exist if we consider all $a\in\Bbb R$. If we restrict ourselves to nonnegative $a,b$, then the minimum is zero. Maybe, may not, the question was about maximum? The minimum depends on the domain of considerations.
A: If $a$ and $b$ are restricted to integers (also known as whole numbers), the only possibilities for $a+b$ you have to check are:
$$\begin{align}
1&+12\\
2&+11\\
3&+10\\
4&+9\\
5&+8\\
6&+7\\
7&+6\\
8&+5\\
9&+4\\
10&+3\\
11&+2\\
12&+1
\end{align}$$
