The selling price of a commodity is fixed at Rs 60 and its cost function is `C(x)=35x+250`
(i) Determine the profit function.
(ii) Find the break even points.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Revenue Function
The selling price of the commodity is given as Rs 60. If we sell \( x \) units, the revenue function \( R(x) \) can be defined as:
\[ R(x) = \text{Selling Price} \times \text{Quantity Sold} = 60x \]

### Step 2: Define the Cost Function
The cost function is provided as:
\[ C(x) = 35x + 250 \]

### Step 3: Determine the Profit Function
The profit function \( P(x) \) is defined as the difference between the revenue function and the cost function:
\[ P(x) = R(x) - C(x) \]
Substituting the expressions for \( R(x) \) and \( C(x) \):
\[ P(x) = 60x - (35x + 250) \]
Simplifying this:
\[ P(x) = 60x - 35x - 250 = 25x - 250 \]

### Step 4: Find the Break-even Points
The break-even point occurs when the profit is zero, i.e., \( P(x) = 0 \):
\[ 25x - 250 = 0 \]
Solving for \( x \):
\[ 25x = 250 \\ x = \frac{250}{25} = 10 \]
Thus, the break-even point is at \( x = 10 \).

### Summary of the Solution
(i) The profit function is:
\[ P(x) = 25x - 250 \]

(ii) The break-even point is:
\[ x = 10 \]