The difference of two numbers is 70. Find the two numbers so their product is as small as
possible.

-35, 35

Step-by-step explanation:

let one of the numbers be x.

It is given that the difference between the 2 numbers is 70. In order for this statement to be true,  the other number must be (x + 70)

so now we have our 2 numbers  x and (x + 70)

let y = the product of our 2 numbers

hence,

y = x (x + 70) , or

y = x² + 70x

We recognize this to be a quadratic equation and we can immediately concluded some things:

1) the x² term is positive, hence it must be a quadratic graph with a minimum   vertex (i.e the graph looks like a u-shape and not an n-shape)

2) the minimum point is located at x = -b/2a.

in our case, b = 70 and a = 1,

hence the for y to be minimum,  

x = -(70) / 2(1) = -70/2

x = -35   (answer)

therefore the 2nd number

= x + 70

= -35 + 70

= 35 (answer)