#### Answer

$(a,b) = (50, -50)$

#### Work Step by Step

Find two numbers whose difference is 100 and whose product is a minimum.
Write out the equations
$a-b=100$ and $y=ab$
Substitute
$a=100+b$
$y=(100+b)b$
$y=100b + b^2$
The problem asks to optimize y, so differentiate and set the derivative $=0$
$y'=100+2b$
$-100=2b$
$b=-50$
Use the second derivative test to confirm that $b=-50$ is in fact a minimum.
$y''= 2$
Concave up on a critical point confirms that $b=-50$ is a minimum.
Therefore, $a=50$.
$(a,b) = (50, -50)$