Michael A. answered • 03/06/17

Tutor

5.0
(1,021)
AFOQT/ASVAB/Accuplacer/TEAS/HESI/SAT/Algebra/Calc I Expert

Let the numbers be x and y.

Then x - y = 40

That means x = 40 + y

Now, let's say F = xy = (40 + y)(y) = y² + 40y

F will have a minimum where its first derivative is either undefined or equal to 0.

F' = 2y + 40 which is continuous and defined for all real numbers y

If 2y + 40 = 0

2y = -40

y = -20

Is -20 a minimum?

Look at a number to the left of -20, say -21. Now, plug that into F'.

F'(-21) = 2(-21) + 40 = -42 + 40 = -2 < 0, so F is decreasing over this interval (-∞, -20)

Now, look at a number to the right of -20. Take -19.

F'(-19) = 2(-19) + 40 = -38 + 40 = 2 > 0

Therefore, F is increasing over the interval (20, ∞)

By the First Derivative Test, -20 is a minimum point of F

Since y = -20, then x = 40 + y = 40 + (-20) = 40 - 20 = 20