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

Answer 1

Maximum means take the derivative so:
P=(−1/10)s^3+6s^2+400dP/ds=(−3/10)s^2+12s
Find the stationary point(s):
0=(−3/10)s^2+12s0=−3s^2+120s0=−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...