The cost, in dollars, for the production of x felt-tipped pens is C(x) = 48 000 + 0.28x + 0.00001x^2, and the revenue is R(x) = 0.68x - 0.00001x^2. What production level maximizes profit?

To find the production level that maximizes profit, we need to determine the profit function and then find its maximum.

The profit function is given by the difference between the revenue function and the cost function:

P(x) = R(x) - C(x)

Substituting the given revenue and cost functions:

P(x) = (0.68x - 0.00001x^2) - (48 000 + 0.28x + 0.00001x^2)

Simplifying, we get:

P(x) = 0.68x - 0.00001x^2 - 48 000 - 0.28x - 0.00001x^2

P(x) = -0.00002x^2 + 0.4x - 48,000

Now, to find the production level that maximizes profit, we need to consider the vertex of the parabolic profit function. The x-coordinate of the vertex gives us the production level that maximizes profit.

To find the x-coordinate of the vertex, we can use the formula:

x = -b/2a

where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c.

In the profit function, we have:

a = -0.00002
b = 0.4

Plugging these values into the formula, we get:

x = -(0.4) / (2 * (-0.00002))

Simplifying further, we get:

x = 10,000

So, the production level that maximizes profit is 10,000 felt-tipped pens.