The profit function, in rupees, of a firm selling t items `(x ge 0)` per week is given by `P(x) =-3500 + (400-x)x`. How many items should the firm sell so that the firm has maximum profit

Options

A. 400
B. 30
C. 200
D. 200

Answer

To find the number of items the firm should sell to maximize profit, we will follow these steps:

### Step 1: Write down the profit function
The profit function is given by:
\[ P(x) = -3500 + (400 - x)x \]

### Step 2: Simplify the profit function
We can expand the profit function:
\[ P(x) = -3500 + 400x - x^2 \]
So, the profit function becomes:
\[ P(x) = -x^2 + 400x - 3500 \]

### Step 3: Differentiate the profit function
To find the maximum profit, we need to differentiate \( P(x) \) with respect to \( x \):
\[ P'(x) = \frac{d}{dx}(-x^2 + 400x - 3500) \]
Using the power rule, we get:
\[ P'(x) = -2x + 400 \]

### Step 4: Set the derivative equal to zero
To find the critical points, we set the derivative equal to zero:
\[ -2x + 400 = 0 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):
\[ 2x = 400 \]
\[ x = 200 \]

### Step 6: Conclusion
The firm should sell **200 items** per week to maximize profit.