The maximum value of the expression `(x^(2)+x+1)/(2x^(2)-x+1)`, for `x in R`, is

Options

A. `(7+2sqrt7)/(7)`
B. `(7-2sqrt7)/(7)`
C. `(7)/(3)`
D. `(14+2sqrt7)/(7)`

Answer

To find the maximum value of the expression \( y = \frac{x^2 + x + 1}{2x^2 - x + 1} \), we will follow these steps:

### Step 1: Cross-multiply
We start by cross-multiplying to eliminate the fraction:
\[ y(2x^2 - x + 1) = x^2 + x + 1 \]
This simplifies to:
\[ 2yx^2 - yx + y = x^2 + x + 1 \]

### Step 2: Rearranging the equation
Rearranging gives us:
\[ (2y - 1)x^2 - (y + 1)x + (y - 1) = 0 \]

### Step 3: Identify coefficients
Now, we can identify the coefficients:
- \( a = 2y - 1 \)
- \( b = -(y + 1) \)
- \( c = y - 1 \)

### Step 4: Condition for real roots
For the quadratic equation to have real roots, the discriminant must be non-negative:
\[ b^2 - 4ac \geq 0 \]
Substituting the coefficients:
\[ (-(y + 1))^2 - 4(2y - 1)(y - 1) \geq 0 \]

### Step 5: Expand the discriminant
Expanding this gives:
\[ (y + 1)^2 - 4(2y^2 - 3y + 1) \geq 0 \]
This simplifies to:
\[ y^2 + 2y + 1 - (8y^2 - 12y + 4) \geq 0 \]
\[ y^2 + 2y + 1 - 8y^2 + 12y - 4 \geq 0 \]
\[ -7y^2 + 14y - 3 \geq 0 \]

### Step 6: Rearranging the inequality
Rearranging gives:
\[ 7y^2 - 14y + 3 \leq 0 \]

### Step 7: Finding roots using the quadratic formula
Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 7 \), \( b = -14 \), and \( c = 3 \):
\[ y = \frac{14 \pm \sqrt{(-14)^2 - 4 \cdot 7 \cdot 3}}{2 \cdot 7} \]
Calculating the discriminant:
\[ y = \frac{14 \pm \sqrt{196 - 84}}{14} \]
\[ y = \frac{14 \pm \sqrt{112}}{14} \]
\[ y = \frac{14 \pm 4\sqrt{7}}{14} \]
\[ y = 1 \pm \frac{2\sqrt{7}}{7} \]

### Step 8: Maximum value
The maximum value occurs at:
\[ y = 1 + \frac{2\sqrt{7}}{7} \]

### Final Result
Thus, the maximum value of the expression \( \frac{x^2 + x + 1}{2x^2 - x + 1} \) is:
\[ \boxed{1 + \frac{2\sqrt{7}}{7}} \]