The cost in millions of dollars for a company to manufacture

The cost in millions of dollars for a company to manufacture x thousand automobiles is given by the function C(X)=3x^2-18x+72. Find the number of automobiles that must be produced to minimize the cost.

Key: 3 thousand automobiles

Please explain (Write work by hand- thank you)

Solution

The cost of manufacuring x thousand automobiles is C(x) = 3x² -18x +72. If C(x) is to be minimum, then dC/dx = 0 and d²C/dx² has to be positive. Here, dC/dx = 6x -18 and d²C/dx² = 6. Thus, if dC/dx = 0, then 6x = 18 or, x = 3. Also, since d²C/dx² is positive, the number of automobiles that must be produced to minimize the cost is 3000.