Final answer:
To find the maximum revenue for R(x) = 2400x - 14x^2 − x^3, calculate its derivative, set it to zero to find critical points, use the second derivative test to identify a maximum, and then substitute that x value into R(x) to find the maximum revenue.
Step-by-step explanation:
To find the level of sales, x, that maximizes revenue for the given total revenue function R(x) = 2400x - 14x2 − x3, we need to find the derivative of the function, which represents the marginal revenue, and then set it to zero to find the critical points. This is often known as finding the turning points of the function where a maximum or minimum could potentially occur.
The derivative of the total revenue function is R'(x) = 2400 - 28x - 3x2. Setting this equal to zero gives us a quadratic equation which we can solve to find the values of x that make the slope of the total revenue function zero. Solving this quadratic equation yields the critical points where the revenue function could be maximized or minimized.
After finding the critical points, we use the second derivative test to determine if those points yield a maximum or a minimum in revenue. The second derivative of the revenue function is R''(x) = -28 - 6x. Substituting the critical points into the second derivative tells us whether the slope is increasing or decreasing at those points, indicating a maximum or minimum revenue respectively.
Once we have identified the critical point that maximizes revenue, we substitute that value of x back into the original revenue function R(x) to find the maximum revenue in dollars. This will give us the highest level of profit, which is the goal of the business.