The daily total cost (in dollars) incurred by Trappee and Sons for producing x cases of TexaPep hot sauce is given by the following function.


c(x)=0.000002x^3+3x+453

(a) Find the average cost function C.
C = ?

(b) Find the level of production that results in the smallest average production cost. (Round your answer to the nearest whole number.)

? cases

(c) Find the level of production for which the average cost is equal to the marginal cost. (Round your answer to the nearest whole number.)

? cases

(a) To find the average cost function C, we need to divide the total cost c(x) by the number of cases x. The formula for average cost is:

C(x) = c(x) / x

For the given function c(x) = 0.000002x^3 + 3x + 453, we substitute this into the formula:

C(x) = (0.000002x^3 + 3x + 453) / x

Simplifying the expression, we have:

C(x) = 0.000002x^2 + 3 + 453/x

Therefore, the average cost function C is:

C(x) = 0.000002x^2 + 3 + 453/x

(b) To find the level of production that results in the smallest average production cost, we need to find the minimum point of the average cost function C(x). This can be done by finding the derivative of C(x) and setting it equal to zero.

Taking the derivative of C(x), we have:

C'(x) = 0.000004x - 453/x^2

Setting C'(x) = 0 and solving for x:

0.000004x - 453/x^2 = 0

0.000004x^3 - 453 = 0

Simplifying further:

x^3 = 113250000

Taking the cube root of both sides:

x = ∛(113250000)

Evaluating the cube root, we find:

x ≈ 487.46

Rounding to the nearest whole number, the level of production that results in the smallest average production cost is:

487 cases

(c) To find the level of production for which the average cost is equal to the marginal cost, we need to equate the average cost function C(x) with the derivative of the cost function c(x).

Setting C(x) equal to the derivative of c(x) and solving for x:

0.000002x^2 + 3 + 453/x = 0.000006x^2 + 3

0.000004x^2 - 453/x = 0

Cross-multiplying and simplifying:

0.000004x^3 - 453 = 0

x^3 = 113250000

Taking the cube root of both sides:

x = ∛(113250000)

Evaluating the cube root, we find:

x ≈ 487.46

Rounding to the nearest whole number, the level of production for which the average cost is equal to the marginal cost is:

487 cases

<vx)-ln[.03]/(l1)<l3>

A shortcut I forgot about until calc III is to make it a function of <vx)
vx=nnx+bbn+c
c(x)=<l.5/.556>
nn(x)
x=nn(.565)
bbn
n=(bb)=.909
c=vvx/.909[pi]^2<nnx>
C=?