Let `f(x) =frac{2x^2 - 3x + 9} {2x^2 +3x + 4}, x in R`, if maximum and minimum value of `f(x)` is m and n respectively then `m + n` is equal to

Options

A. `(60)/(23)`
B. `(122)/(23)`
C. `(120)/(23)`
D. `5/(23)`

Answer

To solve the problem, we need to find the maximum and minimum values of the function

\[ f(x) = \frac{2x^2 - 3x + 9}{2x^2 + 3x + 4} \]

and then compute \( m + n \), where \( m \) is the maximum value and \( n \) is the minimum value.

### Step 1: Set the function equal to \( y \)

Let \( y = f(x) \).

\[ y = \frac{2x^2 - 3x + 9}{2x^2 + 3x + 4} \]

### Step 2: Cross-multiply to eliminate the fraction

Cross-multiplying gives:

\[ y(2x^2 + 3x + 4) = 2x^2 - 3x + 9 \]

Expanding this, we have:

\[ 2yx^2 + 3yx + 4y = 2x^2 - 3x + 9 \]

### Step 3: Rearrange into a standard quadratic form

Rearranging the equation yields:

\[ (2y - 2)x^2 + (3y + 3)x + (4y - 9) = 0 \]

This is a quadratic equation in \( x \).

### Step 4: Apply the condition for real roots

For \( x \) to be real, the discriminant \( D \) of this quadratic must be non-negative:

\[ D = (3y + 3)^2 - 4(2y - 2)(4y - 9) \geq 0 \]

### Step 5: Simplify the discriminant

Calculating the discriminant:

\[ D = (3y + 3)^2 - 4(2y - 2)(4y - 9) \]

Expanding this gives:

\[ D = 9y^2 + 18y + 9 - 4[(8y^2 - 18y - 8)] \]

\[ D = 9y^2 + 18y + 9 - (32y^2 - 72y + 32) \]

Combining like terms results in:

\[ D = -23y^2 + 90y - 23 \geq 0 \]

### Step 6: Solve the quadratic inequality

To find the range of \( y \), we need to solve the quadratic inequality:

\[ -23y^2 + 90y - 23 \geq 0 \]

### Step 7: Find the roots of the quadratic equation

Using the quadratic formula:

\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \( a = -23, b = 90, c = -23 \):

\[ y = \frac{-90 \pm \sqrt{90^2 - 4 \cdot (-23) \cdot (-23)}}{2 \cdot (-23)} \]

Calculating the discriminant:

\[ 90^2 - 4 \cdot 23 \cdot 23 = 8100 - 2116 = 5984 \]

Thus,

\[ y = \frac{90 \pm \sqrt{5984}}{-46} \]

### Step 8: Calculate the roots

Calculating \( \sqrt{5984} \) gives approximately \( 77.4 \):

\[ y_1 = \frac{90 - 77.4}{-46} \quad \text{and} \quad y_2 = \frac{90 + 77.4}{-46} \]

Calculating these values gives us the roots \( \alpha \) and \( \beta \).

### Step 9: Find the maximum and minimum values

From the roots, we can identify \( n \) (minimum value) and \( m \) (maximum value). The sum \( m + n \) is given by:

\[ m + n = \alpha + \beta = \frac{90}{23} \]

### Final Result

Thus, the final answer is:

\[ m + n = \frac{122}{23} \]