`p=(150)/(x^(2)+2)-4` represents the demand function for a product where p is the price per unit for x units. Determine the marginal revenue.

Answer

To determine the marginal revenue from the given demand function \( p = \frac{150}{x^2 + 2} - 4 \), we will follow these steps:

### Step 1: Write the Total Revenue Function
The total revenue \( R \) can be expressed as the product of price \( p \) and quantity \( x \):
\[ R = p \cdot x \]
Substituting the expression for \( p \):
\[ R = \left(\frac{150}{x^2 + 2} - 4\right) \cdot x \]
This simplifies to:
\[ R = \frac{150x}{x^2 + 2} - 4x \]

### Step 2: Differentiate the Total Revenue Function
To find the marginal revenue \( MR \), we need to differentiate the total revenue \( R \) with respect to \( x \):
\[ MR = \frac{dR}{dx} \]
Using the quotient rule for differentiation, where \( u = 150x \) and \( v = x^2 + 2 \):
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Calculating \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
- \( \frac{du}{dx} = 150 \)
- \( \frac{dv}{dx} = 2x \)

Now applying the quotient rule:
\[ MR = \frac{(x^2 + 2)(150) - (150x)(2x)}{(x^2 + 2)^2} \]

### Step 3: Simplify the Expression
Now we simplify the numerator:
\[ MR = \frac{150(x^2 + 2) - 300x^2}{(x^2 + 2)^2} \]
Combining like terms:
\[ MR = \frac{150 \cdot 2 - 150x^2}{(x^2 + 2)^2} \]
This simplifies to:
\[ MR = \frac{300 - 150x^2}{(x^2 + 2)^2} \]

### Final Expression
Thus, the marginal revenue is:
\[ MR = \frac{150(2 - x^2)}{(x^2 + 2)^2} \]

---