To find the maximum revenue for the given revenue function $R(x) = 392x - 0.9x^2$, we can follow these steps:

Step 1: Find the critical points

The critical points occur where the derivative of the function is equal to zero. The derivative of the revenue function is:

$R'(x) = 392 - 1.8x$

Step 2: Set the derivative equal to zero and solve for $x$

$392 - 1.8x = 0$

Solving for $x$:

$1.8x = 392$

$x = \frac{392}{1.8} = 217.78$

Step 3: Verify if it's a maximum

To confirm that this is a maximum, we can check the second derivative of the revenue function:

$R''(x) = -1.8$

Since the second derivative is negative, this confirms that the critical point at $x = 217.78$ gives a maximum.

Step 4: Calculate the maximum revenue

Now, substitute $x = 217.78$ into the original revenue function $R(x) = 392x - 0.9x^2$:

$R(217.78) = 392(217.78) - 0.9(217.78)^2$

Let's calculate this value.The maximum revenue is approximately $R(217.78) = 42684.44$ dollars, rounded to the nearest cent.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the significance of the derivative in finding the maximum of a function?
2. How does the second derivative test confirm if a critical point is a maximum or minimum?
3. Can the revenue function have more than one critical point?
4. How would the result change if the coefficient of $x^2$ in the revenue function were different?
5. What is the economic interpretation of the maximum value of $x$ in this context?

Tip: The second derivative test is a useful method for determining the concavity of a function at critical points, helping identify maxima or minima.