Solution
Step 1:
Write the total cost function
[tex]c(x)\text{ = 0.01x}^3-0.3x^2\text{ + 10x}[/tex]
a) Marginal cost is the first derivative of the total cost
[tex]Marginal\text{ cost c'\lparen x\rparen= 0.03x}^2\text{ - 0.6x + 10}[/tex]
b)
[tex]\begin{gathered} c^{\prime}(x)\text{ = 0.03x}^2\text{ - 0.6x + 10} \\ c^{\prime}(0)\text{ = 0.03 }\times\text{ 0}^2\text{ - 0.6}\times\text{ 0 + 10} \\ c^{\prime}(0)\text{ = 10 additional dollar per item produced} \end{gathered}[/tex]
c)
Graph of the marginal cost
The coordinates of the minimum marginal cost is (10 , 7000)
Minumum marginal cost of 7 additional dollars per item produced occurs when 10 thousand items are produced
d)
[tex]\begin{gathered} 0.03x^2\text{ - 0.6x + 10 = 10} \\ 0.03x^2\text{ - 0.6x = 0} \\ x(0.03x\text{ - 0.6\rparen = 0} \\ \text{x = 0 , x = }\frac{0.6}{0.03} \\ x\text{ = 20} \end{gathered}[/tex]
20 thousands items produced
