f(x)=8/9 x(12-x).

Question

1.                  Reza Cosmetics Company estimates that the number of customers (in thousands) that will use its new product after  months is given by . Estimate the maximum number of customers that will use the product.

Answer 1

f(x)-(8/9)x(12-x)=(8/9)(12x-x^2). If we differentiate we get: f'(x)=(8/9)(12-2x), and when f'(x)=0, we have a maximum or minimum value of f(x). So, when 12-2x=0, 6-x=0 and x=6, and f(x)=(8/9)6(12-6)=32. If we choose values of x near to x=6, say, 5 and 7 we get: f(5)=f(7)=31.11, so f(6)=32 is a maximum, the maximum number of customers.

Another way to solve this is to plot the graph which is a parabola with vertex at (6,32).