Question

What is the minimum product of two numbers whose difference is 30​?

Answer 1

Let's call the two number
`x` and
`y`. Since we know that the difference between them is
`30`, if we assume
`x` to be the largest we have
`x-y=30 \iff x=y+30`

Their product is
`xy`, which we can write as

`(y+30)y = y^2+30y`
This is the equation of a parabola, since it is a polynomial of degree
`2`. To find its minimum, simply derive it and set the derivative to zero: using the power rule

`(d)/(dx)x^n = nx^(n-1)`
we have the following derivative:

`2y+30 = 0 \iff y = -15`
Now that we have found one variable, we can substitute its value in the formula that related it to the other variable:

`x = y+30 = -15+30 = 15`
So, the two numbers are
`-15` and
`15`