If `alpha` is the absolute maximum value of the expression `(3x^(2)+2x-1)/(x^(2)+x+1)AA xx in R`, then `[alpha]` is ___________, (where [.] denotes the greatest integer function)

Answer

To find the absolute maximum value of the expression \( f(x) = \frac{3x^2 + 2x - 1}{x^2 + x + 1} \) for \( x \in \mathbb{R} \), we will follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ f(x) = \frac{3x^2 + 2x - 1}{x^2 + x + 1} \]

### Step 2: Find the Derivative
To find the maximum value, we need to take the derivative of \( f(x) \) and set it to zero. We use the quotient rule:
\[ f'(x) = \frac{(3x^2 + 2x - 1)'(x^2 + x + 1) - (3x^2 + 2x - 1)(x^2 + x + 1)'}{(x^2 + x + 1)^2} \]
Calculating the derivatives:
- \( (3x^2 + 2x - 1)' = 6x + 2 \)
- \( (x^2 + x + 1)' = 2x + 1 \)

Substituting these into the derivative:
\[ f'(x) = \frac{(6x + 2)(x^2 + x + 1) - (3x^2 + 2x - 1)(2x + 1)}{(x^2 + x + 1)^2} \]

### Step 3: Set the Derivative to Zero
Setting the numerator equal to zero:
\[ (6x + 2)(x^2 + x + 1) - (3x^2 + 2x - 1)(2x + 1) = 0 \]

### Step 4: Simplify the Equation
Expand both sides and combine like terms to form a polynomial equation. This will be a cubic equation in \( x \).

### Step 5: Solve for Critical Points
Solve the cubic equation to find the critical points. This may involve factoring or using the cubic formula.

### Step 6: Evaluate \( f(x) \) at Critical Points
Substitute the critical points back into the original function \( f(x) \) to find the corresponding \( f(x) \) values.

### Step 7: Determine the Maximum Value
Compare the values of \( f(x) \) at the critical points and also check the limits as \( x \to \pm \infty \) to ensure we find the absolute maximum.

### Step 8: Find the Greatest Integer Function
Let \( \alpha \) be the absolute maximum value found in the previous step. The final answer will be:
\[ [\alpha] = \text{greatest integer less than or equal to } \alpha \]