Question
The marginal cost of producing x units of an item is \(C'(x)=\frac{1}{4}x-2\)
(A) Find an expression for C(x), assuming the cost of producing 0 units is $2, such that C(0) = 2.
(B) Find the value of x such that the average cost is a minimum. Justify your answer.
(C) Find the cost for producing 40 units.
Answer/Explanation
(A)\(C(x)=\int C'(x)dx=\int \left ( \frac{1}{4}x-2 \right )dx=\frac{1}{8}x^{2}-2x+C=\frac{1}{8}x^{2}-2x+2\)
(B)\(\bar{C}=\frac{C(x)}{x}=\frac{x}{8}-2+\frac{2}{x}.\) Set \(\frac{\mathrm{d} \bar{C}}{\mathrm{d} x}\)equal to zero to find the minimum average cost.\( \frac{\mathrm{d} \bar{C}}{\mathrm{d} x}=\frac{1}{8}-\frac{2}{x^{2}}=0\)Solving for x, one obtains x = 4. Find the second derivative at this point to confirm it’s a relative minimum:\(\frac{\mathrm{d} ^{2}\bar{C}}{\mathrm{d} x^{2}}=\frac{4}{x^{3}}\).At x = 4, this second derivative is positive; therefore, it is concave up and a relative minimum. The value of x such that the average cost is a minimum is 4.
(C)\(C(40)=\frac{40^{2}}{8}-2(40)+2=122\)