If `C=ax^(2)+bx+c` represents the total cost function, then slope of the marginal cost function is

Options

A. a
B. 2a
C. `a/2`
D. b

Answer

To solve the problem, we need to find the slope of the marginal cost function derived from the total cost function given by \( C = ax^2 + bx + c \).

### Step-by-step Solution:

1. **Identify the Total Cost Function**:
The total cost function is given as:
\[ C(x) = ax^2 + bx + c \]

2. **Find the Marginal Cost Function**:
The marginal cost (MC) is defined as the derivative of the total cost function with respect to \( x \):
\[ MC = \frac{dC}{dx} \]

3. **Differentiate the Total Cost Function**:
We will differentiate \( C(x) \):
\[ \frac{dC}{dx} = \frac{d}{dx}(ax^2) + \frac{d}{dx}(bx) + \frac{d}{dx}(c) \]
Using the differentiation rules:
- The derivative of \( ax^2 \) is \( 2ax \).
- The derivative of \( bx \) is \( b \).
- The derivative of a constant \( c \) is \( 0 \).

Therefore, we have:
\[ MC = 2ax + b \]

4. **Find the Slope of the Marginal Cost Function**:
The slope of the marginal cost function is the derivative of the marginal cost with respect to \( x \):
\[ \frac{d(MC)}{dx} = \frac{d}{dx}(2ax + b) \]
Again applying the differentiation rules:
- The derivative of \( 2ax \) is \( 2a \).
- The derivative of \( b \) (a constant) is \( 0 \).

Thus, we find:
\[ \frac{d(MC)}{dx} = 2a \]

5. **Conclusion**:
The slope of the marginal cost function is:
\[ \text{Slope of Marginal Cost} = 2a \]

### Final Answer:
The slope of the marginal cost function is \( 2a \).

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