Answer:
Maximum profits are earned when x = 64 that is when 64 units are sold.
Maximum Profit = P(64) = 2,08,490.666667$
Step-by-step explanation:
We are given the following information:
, where P(x) is the profit function.
We will use double derivative test to find maximum profit.
Differentiating P(x) with respect to x and equating to zero, we get,
![\displaystyle\frac{d(P(x))}{dx} = 6400 - 36x - x^2](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%28P%28x%29%29%7D%7Bdx%7D%20%3D%206400%20-%2036x%20-%20x%5E2)
Equating it to zero we get,
![x^2 + 36x - 6400 = 0](https://tex.z-dn.net/?f=x%5E2%20%2B%2036x%20-%206400%20%3D%200)
We use the quadratic formula to find the values of x:
, where a, b and c are coefficients of
respectively.
Putting these value we get x = -100, 64
Now, again differentiating
![\displaystyle\frac{d^2(P(x))}{dx^2} = -36 - 2x](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%5E2%28P%28x%29%29%7D%7Bdx%5E2%7D%20%3D%20-36%20-%202x)
At x = 64, ![\displaystyle\frac{d^2(P(x))}{dx^2} < 0](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7Bd%5E2%28P%28x%29%29%7D%7Bdx%5E2%7D%20%3C%200)
Hence, maxima occurs at x = 64.
Therefore, maximum profits are earned when x = 64 that is when 64 units are sold.
Maximum Profit = P(64) = 2,08,490.666667$