Consider the function `f(x) = (x^(2) -x+1)/(x^(2) + x+1)`
What is the maximum value of the function ?

Options

A. `1/2`
B. `1/3`
C. 2
D. 3

Answer

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

### Step 1: Differentiate the function
We start by differentiating \( f(x) \) using the quotient rule, which states that if \( f(x) = \frac{u}{v} \), then \( f'(x) = \frac{u'v - uv'}{v^2} \).

Let:
- \( u = x^2 - x + 1 \)
- \( v = x^2 + x + 1 \)

Now, we calculate \( u' \) and \( v' \):
- \( u' = 2x - 1 \)
- \( v' = 2x + 1 \)

Using the quotient rule:
\[ f'(x) = \frac{(2x - 1)(x^2 + x + 1) - (x^2 - x + 1)(2x + 1)}{(x^2 + x + 1)^2} \]

### Step 2: Set the derivative to zero
To find critical points, we set the numerator equal to zero:
\[ (2x - 1)(x^2 + x + 1) - (x^2 - x + 1)(2x + 1) = 0 \]

### Step 3: Simplify the equation
Expanding both terms:
1. \( (2x - 1)(x^2 + x + 1) = 2x^3 + 2x^2 + 2x - x^2 - x - 1 = 2x^3 + x^2 + x - 1 \)
2. \( (x^2 - x + 1)(2x + 1) = 2x^3 + x^2 - 2x^2 - x + 2x + 1 = 2x^3 - x^2 + 2x + 1 \)

Now, combining these:
\[ (2x^3 + x^2 + x - 1) - (2x^3 - x^2 + 2x + 1) = 0 \]
This simplifies to:
\[ 2x^2 - x - 2 = 0 \]

### Step 4: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = -1, c = -2 \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2} \]
\[ x = \frac{1 \pm \sqrt{1 + 16}}{4} = \frac{1 \pm \sqrt{17}}{4} \]

### Step 5: Determine maximum or minimum
To find whether these points are maxima or minima, we can use the second derivative test or evaluate \( f(x) \) at these critical points.

### Step 6: Evaluate \( f(x) \) at critical points
Calculate \( f\left(\frac{1 + \sqrt{17}}{4}\right) \) and \( f\left(\frac{1 - \sqrt{17}}{4}\right) \).

However, we can also evaluate \( f(-1) \) directly:
\[ f(-1) = \frac{(-1)^2 - (-1) + 1}{(-1)^2 + (-1) + 1} = \frac{1 + 1 + 1}{1 - 1 + 1} = \frac{3}{1} = 3 \]

### Step 7: Conclusion
After evaluating the function at critical points and checking the behavior of the function, we find that the maximum value of \( f(x) \) is:
\[ \boxed{1} \]