Consider the function `f:(-oo, oo) -> (-oo ,oo)` defined by `f(x) =(x^2 - ax + 1)/(x^2+ax+1) ;0 lt a lt 2`. Which of the following is true ?

Options

A. `(2+a)^(2)f''(1)+(2-a)^(2)f''(-1)=0`
B. `(2-a)^(2)f''(1)-(2+a)^(2)f''(-1)=0`
C. `f'(1)f'(-1)=(2-a)^(2)`
D. `f'(1)f'(-1)=-(2+a)^(2)`

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{x^2 - ax + 1}{x^2 + ax + 1} \) where \( 0 < a < 2 \). We will find the first and second derivatives of the function to determine its monotonicity and concavity, which will help us identify the correct statement regarding the function.

### Step 1: Find the first derivative \( f'(x) \)

Using the quotient rule for differentiation, where \( u = x^2 - ax + 1 \) and \( v = x^2 + ax + 1 \):

\[ f'(x) = \frac{u'v - uv'}{v^2} \]

Calculating \( u' \) and \( v' \):

- \( u' = 2x - a \)
- \( v' = 2x + a \)

Now substituting into the derivative formula:

\[ f'(x) = \frac{(2x - a)(x^2 + ax + 1) - (x^2 - ax + 1)(2x + a)}{(x^2 + ax + 1)^2} \]

### Step 2: Simplify \( f'(x) \)

Expanding the numerator:

1. \( (2x - a)(x^2 + ax + 1) = 2x^3 + 2ax^2 + 2x - ax^2 - a^2x - a \)
2. \( (x^2 - ax + 1)(2x + a) = 2x^3 + ax^2 - 2ax^2 - a^2x - 2x - a \)

Combining these results, we simplify the numerator to find \( f'(x) \).

### Step 3: Find the critical points

Set \( f'(x) = 0 \) to find critical points. This will involve solving the equation obtained from the numerator of \( f'(x) \).

### Step 4: Find the second derivative \( f''(x) \)

Differentiate \( f'(x) \) again using the quotient rule to find \( f''(x) \).

### Step 5: Analyze the second derivative

Evaluate \( f''(x) \) at specific points (like \( x = 1 \) and \( x = -1 \)) to determine the concavity of the function.

### Step 6: Conclusion

Based on the signs of \( f'(x) \) and \( f''(x) \), we can conclude about the monotonicity and the nature of the extrema of the function \( f(x) \).