For any real value of x, find the minimum value of the rational expression `(x^2+x+1)/(x^2-x+1)`.

Options

A. `-1//6`
B. `1//3`
C. `2//9`
D. 1

Answer

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

### Step 1: Define the function
Let \( f(x) = \frac{x^2 + x + 1}{x^2 - x + 1} \).

### Step 2: Find the derivative
To find the minimum value, we need to differentiate \( f(x) \). We will use the quotient rule for differentiation, which states that if \( f(x) = \frac{g(x)}{h(x)} \), then:
\[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} \]
Here, \( g(x) = x^2 + x + 1 \) and \( h(x) = x^2 - x + 1 \).

Calculating the derivatives:
- \( g'(x) = 2x + 1 \)
- \( h'(x) = 2x - 1 \)

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

### Step 3: Set the derivative to zero
To find critical points, set \( f'(x) = 0 \):
\[ (2x + 1)(x^2 - x + 1) - (x^2 + x + 1)(2x - 1) = 0 \]

### Step 4: 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. \( (x^2 + x + 1)(2x - 1) = 2x^3 + 2x^2 + 2x - x^2 - x - 1 = 2x^3 + x^2 + x - 1 \)

Now, substituting back into the equation:
\[ (2x^3 - x^2 + x + 1) - (2x^3 + x^2 + x - 1) = 0 \]
This simplifies to:
\[ -2x^2 + 2 = 0 \implies x^2 = 1 \implies x = \pm 1 \]

### Step 5: Evaluate the function at critical points
Now, we will evaluate \( f(x) \) at \( x = 1 \) and \( x = -1 \):
- For \( x = 1 \):
\[ f(1) = \frac{1^2 + 1 + 1}{1^2 - 1 + 1} = \frac{3}{1} = 3 \]
- For \( x = -1 \):
\[ f(-1) = \frac{(-1)^2 + (-1) + 1}{(-1)^2 - (-1) + 1} = \frac{1 - 1 + 1}{1 + 1 + 1} = \frac{1}{3} \]

### Step 6: Determine the minimum value
The values we found are:
- \( f(1) = 3 \)
- \( f(-1) = \frac{1}{3} \)

Thus, the minimum value of \( f(x) \) is:
\[ \text{Minimum value} = \frac{1}{3} \]

### Final Answer
The minimum value of the rational expression \( \frac{x^2 + x + 1}{x^2 - x + 1} \) is \( \frac{1}{3} \).