What is the minimum product of $a$ and $b$?

Question

$a$ 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

Answer 1

$b=13-a$

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

$$...$$

Answer 2

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.

Answer 3

If $a$ and $b$ are restricted to integers (also known as whole numbers), the only possibilities for $a+b$ you have to check are: $$\begin{align} 1&+12\\ 2&+11\\ 3&+10\\ 4&+9\\ 5&+8\\ 6&+7\\ 7&+6\\ 8&+5\\ 9&+4\\ 10&+3\\ 11&+2\\ 12&+1 \end{align}$$