A monopolist's demand function is `x = 50 - (P)/4` At what price is marginal revenue zero?

Answer

To find the price at which the marginal revenue is zero for the given monopolist's demand function, we can follow these steps:

### Step 1: Write down the demand function
The demand function is given as:
\[ x = 50 - \frac{P}{4} \]

### Step 2: Rearrange the demand function to express price (P) in terms of quantity (x)
To find the price as a function of quantity, we can rearrange the equation:
\[ \frac{P}{4} = 50 - x \]
Multiplying both sides by 4 gives:
\[ P = 200 - 4x \]

### Step 3: Write down the revenue function (R)
The revenue (R) is given by the product of price (P) and quantity (x):
\[ R = P \cdot x \]
Substituting the expression for P from Step 2:
\[ R = (200 - 4x) \cdot x \]
Expanding this, we get:
\[ R = 200x - 4x^2 \]

### Step 4: Find the marginal revenue (MR)
Marginal revenue is the derivative of the revenue function with respect to quantity (x):
\[ MR = \frac{dR}{dx} \]
Differentiating the revenue function:
\[ MR = \frac{d}{dx}(200x - 4x^2) \]
Using the power rule of differentiation:
\[ MR = 200 - 8x \]

### Step 5: Set marginal revenue to zero and solve for x
To find the quantity at which marginal revenue is zero, we set the MR equation to zero:
\[ 200 - 8x = 0 \]
Solving for x:
\[ 8x = 200 \]
\[ x = \frac{200}{8} = 25 \]

### Step 6: Substitute x back into the price equation to find P
Now that we have the quantity (x = 25), we can substitute it back into the price equation:
\[ P = 200 - 4x \]
Substituting x:
\[ P = 200 - 4(25) \]
\[ P = 200 - 100 = 100 \]

### Final Answer
The price at which marginal revenue is zero is:
\[ P = 100 \]

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