The demand function of a monopolist is p = 300 - 15 x .Find the price at which the mar-ginal revenue vanishes.

Answer

To solve the problem, we need to find the price at which the marginal revenue (MR) of the monopolist vanishes. We start with the given demand function and follow these steps:

### Step 1: Write down the demand function
The demand function given is:
\[ P = 300 - 15x \]

### Step 2: Calculate Total Revenue (TR)
Total Revenue (TR) is calculated as the product of price (P) and quantity (x):
\[ TR = P \cdot x = (300 - 15x) \cdot x \]
\[ TR = 300x - 15x^2 \]

### Step 3: Find Marginal Revenue (MR)
Marginal Revenue (MR) is the derivative of Total Revenue (TR) with respect to quantity (x):
\[ MR = \frac{d(TR)}{dx} = \frac{d(300x - 15x^2)}{dx} \]

Differentiating each term:
- The derivative of \( 300x \) is \( 300 \).
- The derivative of \( -15x^2 \) is \( -30x \).

So, we have:
\[ MR = 300 - 30x \]

### Step 4: Set Marginal Revenue to zero
To find the quantity (x) at which MR vanishes, we set MR equal to zero:
\[ 300 - 30x = 0 \]

### Step 5: Solve for x
Rearranging the equation:
\[ 30x = 300 \]
\[ x = \frac{300}{30} \]
\[ x = 10 \]

### Step 6: Find the price at x = 10
Now, we substitute \( x = 10 \) back into the demand function to find the price (P):
\[ P = 300 - 15(10) \]
\[ P = 300 - 150 \]
\[ P = 150 \]

### Final Answer
The price at which the marginal revenue vanishes is:
\[ \boxed{150} \]