The total cost function is given by C(x) =` 2x^(3) - 3 . 5 x^(2) + x` . The point where MC curve cuts y-axis is

Options

A. (1,0)
B. (0,0)
C. (0,1)
D. `(0,(1)/(6))`

Answer

To find the point where the Marginal Cost (MC) curve cuts the y-axis for the given total cost function \( C(x) = 2x^3 - 3.5x^2 + x \), we will follow these steps:

### Step 1: Differentiate the Total Cost Function
To find the Marginal Cost, we need to differentiate the total cost function \( C(x) \) with respect to \( x \).

\[ MC(x) = \frac{dC}{dx} = \frac{d}{dx}(2x^3 - 3.5x^2 + x) \]

### Step 2: Apply the Power Rule
Using the power rule of differentiation, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \), we differentiate each term:

- The derivative of \( 2x^3 \) is \( 6x^2 \).
- The derivative of \( -3.5x^2 \) is \( -7x \).
- The derivative of \( x \) is \( 1 \).

Thus, we have:

\[ MC(x) = 6x^2 - 7x + 1 \]

### Step 3: Evaluate Marginal Cost at the y-axis
The y-axis corresponds to \( x = 0 \). We will substitute \( x = 0 \) into the Marginal Cost function:

\[ MC(0) = 6(0)^2 - 7(0) + 1 \]

### Step 4: Simplify the Expression
Now we simplify the expression:

\[ MC(0) = 0 - 0 + 1 = 1 \]

### Step 5: Identify the Point
The point where the Marginal Cost curve cuts the y-axis is given by the coordinates \( (0, MC(0)) \). Therefore, the point is:

\[ (0, 1) \]

### Conclusion
The point where the MC curve cuts the y-axis is \( (0, 1) \).

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