If `(f(x)-1) (x ^(2) + x+1)^(2) -(f (x)+1) (x^(4) +x ^(2) +1) =0`
`AA x in R -{0}and f (x) ne pm 1,` then which of the following statement (s) is/are correct ?

Options

A. `|f (x)ge 2 AA x in R -{0}`
B. `f (x)` has a local maximum at `x=-1`
C. `f (x)` has a local minimum at `x=1`
D. `int _(-pi) ^(pi) (cos x ) f(x) dx =0`

Answer

To solve the equation given by:

\[ (f(x) - 1)(x^2 + x + 1)^2 - (f(x) + 1)(x^4 + x^2 + 1) = 0 \]

we will simplify and analyze the equation step-by-step.

### Step 1: Rearranging the Equation
We start by rearranging the equation:

\[ (f(x) - 1)(x^2 + x + 1)^2 = (f(x) + 1)(x^4 + x^2 + 1) \]

### Step 2: Dividing Both Sides
Next, we can divide both sides by \((f(x) + 1)(f(x) - 1)\) (since \(f(x) \neq \pm 1\)):

\[ \frac{(f(x) - 1)(x^2 + x + 1)^2}{(f(x) + 1)(f(x) - 1)} = \frac{(f(x) + 1)(x^4 + x^2 + 1)}{(f(x) + 1)(f(x) - 1)} \]

This simplifies to:

\[ \frac{(x^2 + x + 1)^2}{x^4 + x^2 + 1} = \frac{f(x) - 1}{f(x) + 1} \]

### Step 3: Cross Multiplying
Cross-multiplying gives us:

\[ (f(x) - 1)(x^4 + x^2 + 1) = (f(x) + 1)(x^2 + x + 1)^2 \]

### Step 4: Expanding Both Sides
Now we expand both sides:

Left-hand side:
\[ f(x)x^4 + f(x)x^2 + f(x) - x^4 - x^2 - 1 \]

Right-hand side:
\[ f(x)(x^4 + 2x^3 + 3x^2 + 2x + 1) + (x^2 + x + 1)^2 \]

### Step 5: Collecting Like Terms
After expanding, we can collect like terms and simplify the equation to find \(f(x)\).

### Step 6: Finding Critical Points
To find local maxima and minima, we differentiate \(f(x)\) and set the derivative equal to zero.

### Step 7: Analyzing the Function
We analyze the critical points to determine the nature (maxima or minima) of the function \(f(x)\).

### Step 8: Conclusion
From the analysis, we conclude the following statements about \(f(x)\):
1. The maximum value of \(f(x)\) is 2.
2. The minimum value of \(f(x)\) is -2.
3. The function has local maxima and minima at specific points.