Find the maximum revenue for the revenue function R(x)=392x-0.9x^2. R=$
Round the nearest cent

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.

Find the maximum revenue for the revenue function R(x)=392x-0.9x^2. R=$. Round the nearest cent.

Discover how to find the maximum revenue for the quadratic function R(x) = 392x - 0.9x^2 using calculus. This guide includes critical points, derivative tests, and rounding to the nearest cent, perfect for students studying calculus at the high school or college level.

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