If the marginal revenue function of a commodity is `MR=2x-9x^(2)` then the revenue function is

Options

A. `2x^(2)-9x^(3)`
B. `2-18x`
C. `x^(2)-3x^(3)`
D. `18+x^(2)-3x^(3)`

Answer

To find the revenue function from the given marginal revenue function \( MR = 2x - 9x^2 \), we will integrate the marginal revenue function. Here are the steps:

### Step 1: Write down the marginal revenue function
The marginal revenue function is given by:
\[ MR = 2x - 9x^2 \]

### Step 2: Integrate the marginal revenue function
To find the revenue function \( R(x) \), we need to integrate \( MR \):
\[ R(x) = \int (2x - 9x^2) \, dx \]

### Step 3: Perform the integration
Now we will integrate each term separately:
\[ R(x) = \int 2x \, dx - \int 9x^2 \, dx \]
\[ = 2 \cdot \frac{x^2}{2} - 9 \cdot \frac{x^3}{3} + C \]
\[ = x^2 - 3x^3 + C \]

### Step 4: Determine the constant of integration
To find the constant \( C \), we can use the information that when \( x = 0 \), the revenue \( R(0) \) should also be 0:
\[ R(0) = 0^2 - 3(0)^3 + C = 0 \]
This implies:
\[ C = 0 \]

### Step 5: Write the final revenue function
Substituting \( C = 0 \) back into the equation, we have:
\[ R(x) = x^2 - 3x^3 \]

### Conclusion
Thus, the revenue function is:
\[ R(x) = x^2 - 3x^3 \]