Respuesta :
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:[tex]P(x) = 6400x - 18x^2 - \frac{x^3}{3} - 40000[/tex], 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,
[tex]\displaystyle\frac{d(P(x))}{dx} = 6400 - 36x - x^2[/tex]
Equating it to zero we get,
[tex]x^2 + 36x - 6400 = 0[/tex]
We use the quadratic formula to find the values of x:
[tex]x = \displaystyle\frac{-b \pm \sqrt{b^2 - 4ac} }{2a}[/tex], where a, b and c are coefficients of [tex]x^2, x^1 , x^0[/tex] respectively.
Putting these value we get x = -100, 64
Now, again differentiating
[tex]\displaystyle\frac{d^2(P(x))}{dx^2} = -36 - 2x[/tex]
At x = 64, [tex]\displaystyle\frac{d^2(P(x))}{dx^2} < 0[/tex]
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$
