Functions: find number t-shirts and cost

Question

A company that makes and sells t-shirts has found that the total monthly cost C in dollars of producing x shirts is given by the function C(x) = 1/6 x^2 - 50x + 4500. Find the number of t-shirts that should be produced to minimize the monthly cost and state what the cost will be.

Answer 1

C(x) = x^2/6 -50x +4500

take the derivative and set equal to zero

C'(x) = x/3 -50 = 0

x -150 = 0

x =150 shirts per month

C(150) = (1/6)(150^2) -50(150) +4500

= 22,500/6 - 7500+4500

= 3750-3000

=$750 = minimum cost per month

graphically, the cost function is an upward opening parabola with vertex= minimum point = (150, 750)