Question

A cosmetics company is planning the introduction and promotion of a new lipstick line. The marketing research department has found that the demand in a particular city is given approximately by p=8ex,0≤x≤2, where x thousand lipsticks were sold per week at a price of p dollars each. At what price will the weekly revenue be maximized?

Answer 1

Answer:

p = 59.11 dollars

Explanation:

Given

Price:     p(x) = 8eˣ      (0 ≤ x ≤ 2)

Revenue;  R = x*p = 8xeˣ

p = ?  when R be at maximum

We can apply

dR/dx = d(x*p)/dx = 0

⇒  d(8xeˣ)/dx = 8*(1*eˣ + x*eˣ) = 0

⇒  eˣ*(1 + x) = 0    ⇒    x = - 1

as x = - 1 ∉ [0, 2]

then, we have

p(0) = 8e⁰ = 8

R = 0*8 = 0

If x = 1

p(1) = 8e¹ ≈ 21.74

R = 1*21.74 = 21.74

If x = 2

p(2) = 8e² ≈ 59.11

R = 2*59.11 = 118.22

Implies that, R(x) is maximum at x = 2.

   

Thus, the price that maximize the revenue of the company is 59.11 dollars.