If the total cost function of producing x units of a commodity is given by `360 – 12x +2x^(2)`, then the level of output at which the total cost is minimum is

Options

A. 24
B. 12
C. 6
D. 3

Answer

To find the level of output at which the total cost is minimized, we will follow these steps:

### Step 1: Write down the total cost function.
The total cost function is given by:
\[ C(x) = 360 - 12x + 2x^2 \]

### Step 2: Differentiate the total cost function with respect to \( x \).
To find the critical points, we need to differentiate the cost function:
\[ \frac{dC}{dx} = \frac{d}{dx}(360 - 12x + 2x^2) \]

Using the rules of differentiation:
- The derivative of a constant (360) is 0.
- The derivative of \(-12x\) is \(-12\).
- The derivative of \(2x^2\) is \(4x\) (using the power rule).

Thus, we have:
\[ \frac{dC}{dx} = 0 - 12 + 4x = 4x - 12 \]

### Step 3: Set the derivative equal to zero to find critical points.
To find the minimum cost, we set the derivative equal to zero:
\[ 4x - 12 = 0 \]

### Step 4: Solve for \( x \).
Now, solve for \( x \):
\[ 4x = 12 \]
\[ x = \frac{12}{4} = 3 \]

### Step 5: Verify that this point is a minimum.
To confirm that this point corresponds to a minimum, we can check the second derivative:
\[ \frac{d^2C}{dx^2} = \frac{d}{dx}(4x - 12) = 4 \]

Since the second derivative is positive (\(4 > 0\)), this indicates that the function is concave up at \( x = 3 \), confirming that it is indeed a minimum.

### Conclusion:
The level of output at which the total cost is minimized is:
\[ x = 3 \]