The average cost function, AC for a commodity is given by ` AC = x + 5+ (36)/(x)`, in terms of output x.
Find :
(i) The total cost, C and marginal cost, MC as a function of x.
(ii) The outputs for which AC increases.

Answer

To solve the problem, we will follow the steps outlined in the video transcript and provide a detailed solution.

### Step 1: Find the Total Cost Function (C)

The average cost (AC) function is given by:

\[ AC = x + 5 + \frac{36}{x} \]

To find the total cost (C), we use the relationship between average cost and total cost:

\[ C = AC \times x \]

Substituting the expression for AC into this equation, we get:

\[ C = \left(x + 5 + \frac{36}{x}\right) \times x \]

Now, we distribute \(x\) across the terms in the parentheses:

\[ C = x^2 + 5x + 36 \]

### Step 2: Find the Marginal Cost (MC)

The marginal cost (MC) is the derivative of the total cost (C) with respect to output (x):

\[ MC = \frac{dC}{dx} \]

Now, we differentiate the total cost function:

\[ C = x^2 + 5x + 36 \]

Taking the derivative:

\[ MC = \frac{d}{dx}(x^2 + 5x + 36) = 2x + 5 \]

### Step 3: Find the Outputs for Which AC Increases

To determine when the average cost (AC) increases, we need to find the derivative of the AC function and set it greater than zero:

\[ AC = x + 5 + \frac{36}{x} \]

Taking the derivative of AC:

\[ \frac{d(AC)}{dx} = 1 - \frac{36}{x^2} \]

Setting the derivative greater than zero for AC to be increasing:

\[ 1 - \frac{36}{x^2} > 0 \]

Rearranging the inequality:

\[ 1 > \frac{36}{x^2} \]

Multiplying both sides by \(x^2\) (assuming \(x > 0\)):

\[ x^2 > 36 \]

Taking the square root of both sides:

\[ x > 6 \quad \text{or} \quad x < -6 \]

Since \(x\) represents output and must be positive, we conclude:

\[ x > 6 \]

### Final Answers

(i) The total cost function \(C\) is:

\[ C = x^2 + 5x + 36 \]

The marginal cost function \(MC\) is:

\[ MC = 2x + 5 \]

(ii) The outputs for which the average cost increases are:

\[ x > 6 \]