# a company makes x toys daily at a cost of C(x)=125+30x+2(x to the power of 2/3) dollars. what daily production level will minimize the average cost? (note define average cost as the Total cost divided by the total number of items)

thx

According to your definition
Average Cost = C(x)/x
=125/x + 30 + 2x^(-1/3)

d(Average Cost)/dx = -125/x^2 - (2/3)x^(-4/3)

set this to zero and solve

(I got x=32.75, but x should be a whole number, so x = 33)

## To minimize the average cost, we need to find the production level at which the derivative of the average cost with respect to the number of items (x) is equal to zero. This is because at the minimum point, the slope of the average cost function is zero.

Let's start by calculating the derivative of the average cost function:

d(Average Cost)/dx = -125/x^2 - (2/3)x^(-4/3)

To find where this derivative equals zero, we can set it equal to zero and solve for x:

-125/x^2 - (2/3)x^(-4/3) = 0

Multiplying both sides by x^2 to get rid of the denominators:

-125 - (2/3)x^(2/3) = 0

Rearranging the equation:

(2/3)x^(2/3) = -125

Multiplying both sides by 3/2:

x^(2/3) = -125 * (3/2)

x^(2/3) = -187.5

To isolate x, we need to raise both sides of the equation to the power of (3/2):

(x^(2/3))^(3/2) = (-187.5)^(3/2)

x = (-187.5)^(3/2)

Now, solving for x will give us the production level that minimizes the average cost. However, (-187.5)^(3/2) is a negative value, which is not practical in this context. Since the production level should be a whole number, we can round the result up or down to the nearest integer.

In this case, rounding up to the nearest whole number gives us x = 33 as the production level that minimizes the average cost.