Answer:
The producer will have maximum profit when x is 5000.
Step-by-step explanation:
Given C(x) = 0.000002x³ - 0.03x² + 400x + 80000
To find the level of production that will yield the maximum profit for the manufacturer, firstly, we differentiate C(x) to obtain C'(x), we then set the resulting quadratic expression to zero and solve. Whichever of the two values obtained from the quadratic equation will be tested to see what we want to find.
Differentiate C(x)
C'(x) = 3(0.000002)x² - 2(0.03)x + 400
= 0.000006x² - 0.06x + 400
Set C'(x) = 0
0.000006x² - 0.06x + 400 = 0
Solve using quadratic formula.
x = [-b ± √(b² - 4ac)]/2a
a = 0.000006
b = -0.06
c = 400
x = [-(-0.06) ± √((-0.06)² - 4(0.000006 × 400)]/2(0.000006)
= [0.06 ± √(0.0036 - 0.0096]/0.000012
= (0.06/0.000012) ± i√(0.006)/0.000012
= 5000 ± 6454.97i
x = 5000 + 6454.97i
Or
x = 5000 - 6454.97i
