Local Maximum Profit of a Beverage Company
(a) During what year did a local maximum profit occur?
A local extremum occurs at a critical number off. Because the derivative exists for every x, the only critical number(s) occur where the derivative is zero. Find the derivative of f(x).
(b) What was the maximum profit?
A local maximum profit occurred in the year
Answer:
A local maximum profit occurred in the year 2020. The maximum profit was $1150 million.
To find the year and value of the local maximum profit, we need to find the critical points of the profit function. Critical points occur where the derivative of the function is zero. Let's find the derivative of the profit function:
f'(x) = -60x + 660
Now, we set the derivative equal to zero and solve for x:
-60x + 660 = 0
-60x = -660
x = -660 / -60
x = 11
So, the critical point occurs at x = 11. To find the corresponding year, we need to add 9 to x:
Year = x + 9 = 11 + 9 = 20
Therefore, a local maximum profit occurred in the year 2020.
To find the maximum profit value, we substitute the x-value of the critical point back into the original profit function:
f(11) = -30(11)² + 660(11) - 2480
f(11) = -30(121) + 7260 - 2480
f(11) = -3630 + 7260 - 2480
f(11) = 1150
Therefore, the maximum profit was $1150 million.