The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars on advertising is

P= -(1/10)s^3 + 6s^2 + 400

Find the amount of money the company should spend on advertising in order to yield a maximum profit

Question

The profit P (in thousands of dollars) for a company spending an amount s (in thousands of dollars on advertising is

P= -(1/10)s^3 + 6s^2 + 400

Find the amount of money the company should spend on advertising in order to yield a maximum profit

anonymous · Answer

Maximum means take the derivative so:

$$P = (\frac{-1}{10})s^{3} + 6s^{2} + 400$$$$\frac{dP}{ds} = (\frac{-3}{10})s^{2} +12s$$
Find the stationary point(s):

$$0 = (\frac{-3}{10})s^{2} +12s$$$$0 = -3s^{2} + 120s$$$$0 = -3s(s - 40)$$
So our stationary points are 0 and 40. Going back to the original formula, zero advertising gives us a profit of $400K (read the instructions carefully to understand why it's 400,000) while $40K in advertising gives us a profit of -64,000 + 9600 + 400 = -$54,000K so, surprisingly, the company should not spend anything on advertising!

anonymous · Answer

OHHH OK that's what my teacher got but he didn't explain very well and the next question asks about the point of diminishing returns, is that just once the profit reaches the top it starts going down for the company?

anonymous · Answer

Exactly. You keep investing more time/money/resources and you start getting less profit back instead of more.

anonymous · Answer

Thanks so much!! you've really helped! :D