Answer:
- The number of t-shirts he needs to print to obtain maximum profit = 2.79 (in thousand), that is, 2790 t-shirts.
- The maximum profit for this number of shirts is then = 12.208761 (in thousand dollars) = $12209
Step-by-step explanation:
Complete Question
Keegan is printing and selling his original design on t-shirts. He has concluded that for x shirts, in thousands sold his total profits will be p(x) = -x³ + 4x² + x dollars, in thousands will be earned. How many t-shirts (rounded to the nearest whole number) should he print in order to make maximum profits? What will his profits rounded to the nearest whole dollar be if he prints that number of shirts?
The profit function is given as
p(x) = -x³ + 4x² + x
The maximum profit will be obtained by investigating the maximum value of the profit function
At the maximum value of the function,
(dp/dx) = 0 and (d²p/dx²) < 0
p(x) = -x³ + 4x² + x
(dp/dx) = -3x² + 8x + 1
at maximum point
(dp/dx) = -3x² + 8x + 1 = 0
Solving the quadratic equation
x = -0.12 or 2.79
(d²p/dx²) = -6x + 8
at x = -0.12
(d²p/dx²) = -6(0.12) + 8 = 7.28 > 0 (not a maximum point)
At x = 2.79
(d²p/dx²) = -6(2.79) + 8 = -8.74 < 0 (this corresponds to a maximum point!)
So, the maximum of the profit function exists when the number of shirts, x = 2.79 (in thousand).
So, the maximum profits that corresponds to this number of t-shirts is obtained from the profit function.
p(x) = -x³ + 4x² + x
p(x) = -(2.79)³ + 4(2.79²) + 2.79
p(x) = -21.717639 + 31.1364 + 2.79
p(x) = 12.208761 (in thousand dollars) = $12209 to the mearest whole number.
Hope this Helps!!!
