C (x) = 0.000002x3 - 0.03x2 + 400x + 80,000 where C (x) denotes the total cost incurred in producing x sets. Find the level of production that will yield a maximum profit for the manufacturer. Hint: Use the quadratic formula

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