If `C(x)=3x ((x+7)/(x+5))+5` is the total cost of production of x units a certain product, then then marginal cost

Options

A. A) Falls continuously as the output x increases
B. B) Increases continuously as the output x increase
C. C) Neither increases nor decreases as the output x increases
D. D) None of these

Answer

To find the marginal cost from the given total cost function \( C(x) = 3x \left( \frac{x+7}{x+5} \right) + 5 \), we need to differentiate the cost function with respect to \( x \).

### Step-by-step Solution:

1. **Write down the total cost function:**
\[ C(x) = 3x \left( \frac{x+7}{x+5} \right) + 5 \]

2. **Differentiate \( C(x) \) with respect to \( x \):**
To find the marginal cost, we need to compute \( C'(x) \). We will use the product rule for differentiation since \( C(x) \) has a product of two functions:
\[ C'(x) = \frac{d}{dx} \left[ 3x \cdot \frac{x+7}{x+5} \right] + \frac{d}{dx}(5) \]
The derivative of a constant (5) is 0, so we focus on differentiating the product.

3. **Apply the product rule:**
Let \( u = 3x \) and \( v = \frac{x+7}{x+5} \).
The product rule states that \( (uv)' = u'v + uv' \).

- Differentiate \( u \):
\[ u' = 3 \]

- Differentiate \( v \) using the quotient rule:
\[ v = \frac{x+7}{x+5} \quad \Rightarrow \quad v' = \frac{(x+5)(1) - (x+7)(1)}{(x+5)^2} = \frac{x + 5 - x - 7}{(x+5)^2} = \frac{-2}{(x+5)^2} \]

4. **Combine using the product rule:**
\[ C'(x) = 3 \cdot \frac{x+7}{x+5} + 3x \cdot \frac{-2}{(x+5)^2} \]

5. **Simplify the expression:**
\[ C'(x) = 3 \cdot \frac{x+7}{x+5} - \frac{6x}{(x+5)^2} \]
To combine these fractions, find a common denominator:
\[ C'(x) = \frac{3(x+7)(x+5) - 6x}{(x+5)^2} \]

6. **Expand and simplify the numerator:**
\[ 3(x+7)(x+5) = 3(x^2 + 12x + 35) = 3x^2 + 36x + 105 \]
Therefore,
\[ C'(x) = \frac{3x^2 + 36x + 105 - 6x}{(x+5)^2} = \frac{3x^2 + 30x + 105}{(x+5)^2} \]

7. **Final expression for marginal cost:**
\[ C'(x) = \frac{3x^2 + 30x + 105}{(x+5)^2} \]

### Conclusion:
The marginal cost \( C'(x) \) is given by:
\[ C'(x) = \frac{3x^2 + 30x + 105}{(x+5)^2} \]