Question

If p(x) = 3+4x-4x^2 represents the profits in selling x thousand boombotix speakers, how many speakers should be sold to maximize profits?

Answer 1

Hello from MrBillDoesMath!

Answer: 500

Discussion:

The idea is to represent p(x) with a squared component...

3 + 4x - 4x^2 =

3 - ( 4x^2 - 4x) =

Add and subtract 1:

(3+1) - (4x^2-4x +1) =

4 - (2x-1)^2

To maximize profit we need to minimize the term (2x-1)^2 as it is always positive (or zero) and subtracts from the number 4. The minimum of (2x -1)^2 occurs when 2x -1 = 0 or x = 1/2. Since x is the number of speakers in thousands, (1/2)x = 500 speakers.

Note: this problem can be more easily addressed in a Calculus course by taking the first derivative of p(x), setting it to zero, and solving for x (which yields x = 1/2 too)

Thank you,

MrB