The marginal cost, marginal revenue, and marginal profit are derived from the cost and revenue functions using calculus, but the idea is that they represent the cost, revenue, and profit of producing or selling one more unit.
Given the cost function C(x) = 2x, the marginal cost does not change with respect to the number of units produced and is a constant, 2.
The revenue function given is R(x) = 6x - 0.001x^2. The derivative for revenue represents marginal revenue. When you calculate it, marginal revenue will be MR(x) = 6 - 0.002x.
The Marginal Profit is the derivative of the profit function (Profit(x) = Revenue(x) - Cost(x)). It can also be calculated as the difference between Marginal Revenue and Marginal Cost. So, the Marginal Profit function is MP(x) = MR(x) - MC(x) = (6 - 0.002x) - 2.
So, based on the definitions and the given cost and revenue functions, the Marginal Cost is 2, the Marginal Revenue is 6 - 0.002x, and the Marginal Profit is 4 - 0.002x.
Now we want to find all values of x for which the marginal profit is zero.
To find this, we set the marginal profit equation to zero and solve for x:
0 = 4 - 0.002x. Solving this equation will give you x = 2000. This means that the marginal profit will be zero when we produce or sell 2000 units. The marginal cost and marginal revenue are equal at this level of production or sales, so this would typically be the most profitable level to produce or sell.