Given that the marginal cost MC and average cost AC for a product are equal. Then the total cost C is a (i) constant function (ii) linear function of number of units (x) produced (iii) quadratic function of number of units (x) produced (iv) None of these

Options

A. constant function
B. linear function of number of units (x) produced
C. quadratic function of number of units (x) produced
D. None of these

Answer

To solve the problem, we need to analyze the relationship between marginal cost (MC), average cost (AC), and total cost (C). Let's break it down step by step:

### Step 1: Understand the relationship between MC and AC
Given that:
\[ MC = AC \]

### Step 2: Express Average Cost in terms of Total Cost
The average cost (AC) is defined as:
\[ AC = \frac{C}{x} \]
where \( C \) is the total cost and \( x \) is the number of units produced.

### Step 3: Set up the equation
Since we know that \( MC = AC \), we can write:
\[ MC = \frac{C}{x} \]

### Step 4: Express Marginal Cost
The marginal cost (MC) is defined as the derivative of total cost with respect to the number of units produced:
\[ MC = \frac{dC}{dx} \]

### Step 5: Substitute the expression for MC
From the previous steps, we have:
\[ \frac{dC}{dx} = \frac{C}{x} \]

### Step 6: Rearranging the equation
We can rearrange this equation as follows:
\[ dC = \frac{C}{x} dx \]

### Step 7: Separate variables
We can separate the variables to integrate:
\[ \frac{dC}{C} = \frac{dx}{x} \]

### Step 8: Integrate both sides
Now, we integrate both sides:
\[ \int \frac{dC}{C} = \int \frac{dx}{x} \]

This gives us:
\[ \ln |C| = \ln |x| + C_1 \]
where \( C_1 \) is the constant of integration.

### Step 9: Exponentiate to solve for C
Exponentiating both sides, we have:
\[ C = e^{C_1} \cdot x \]
Let \( A = e^{C_1} \), then:
\[ C = A \cdot x \]

### Step 10: Conclusion about the total cost
The equation \( C = A \cdot x \) represents a linear function of \( x \).

### Final Answer
Thus, the total cost \( C \) is a linear function of the number of units \( x \) produced. The correct option is:
(ii) linear function of number of units (x) produced.

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