suppose that c(x)=3x^3 - 12x^2 +9000x is the cost of manufacturing x items. find a production level that will minimize the average cost of making x items
c(x)=3x^3 - 12x^2 +9000x
c ' (x) = 9x^2 - 24x + 9000
= 0 for a max/min of c(x)
9x^2 - 24x + 9000 = 0
This equation has no real solution.
c(0) = 0, for all values of x > 0, c(x) increases
so our minimum cost is when we don't produce anything, (that actually makes sense), and the cost increases rapidly the more we produce
---> not a good business practise.
check your equation
To find the production level that minimizes the average cost of making x items, we need to find the minimum point of the average cost function.
The average cost function can be determined by dividing the total cost function, C(x), by the number of items produced, x. Therefore, the average cost function (AC) is given by:
AC(x) = C(x) / x
Substituting the given cost function, c(x) = 3x^3 - 12x^2 + 9000x, into the average cost formula, we get:
AC(x) = (3x^3 - 12x^2 + 9000x) / x
Simplifying the expression by canceling out the x terms in the numerator, we obtain:
AC(x) = 3x^2 - 12x + 9000
Now, we need to find the value of x that minimizes the average cost. To do this, we take the derivative of the average cost function with respect to x and set it equal to zero. The critical points will tell us where the average cost function reaches a minimum.
Differentiating AC(x) with respect to x, we get:
AC'(x) = 6x - 12
To find the critical points, set AC'(x) equal to zero and solve for x:
6x - 12 = 0
6x = 12
x = 2
Hence, x = 2 is the production level that will minimize the average cost of making x items.