If the revenue function is R(x) = 20x - 0.5x^2, find the quantity that maximizes

If the revenue function is R(x) = 20x - 0.5x^2, find the quantity that maximizes revenue. (2021)

If the revenue function is R(x) = 20x - 0.5x^2, find the quantity that maximizes revenue. (2021)

Step 1: Understand the revenue function R(x) = 20x - 0.5x^2. This function tells us how much money we make based on the quantity x sold.
Step 2: To find the quantity that maximizes revenue, we need to find the derivative of the revenue function, which is R'(x).
Step 3: Calculate the derivative R'(x). The derivative of R(x) = 20x - 0.5x^2 is R'(x) = 20 - x.
Step 4: Set the derivative equal to zero to find the critical points: 20 - x = 0.
Step 5: Solve for x. Adding x to both sides gives us 20 = x, so x = 20.
Step 6: This value of x (20) is the quantity that maximizes revenue.

Revenue Maximization – Understanding how to find the quantity that maximizes revenue using calculus, specifically by taking the derivative of the revenue function and setting it to zero.
Critical Points – Identifying critical points from the first derivative to determine maximum or minimum values.
Second Derivative Test – Using the second derivative to confirm whether the critical point found is a maximum or minimum.