If f(x) =` (1 - x +x^2)/(1+x+x^2) le 3 AA x in R ` then max. value of. ` (1 +2x +4x^2)/(1-2x+4x^2)` is

Options

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

Answer

To solve the problem, we need to analyze the function given by:

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

and find the maximum value of:

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

under the constraint that \( f(x) \leq 3 \) for all \( x \in \mathbb{R} \).

### Step 1: Analyze the function \( f(x) \)

We start by simplifying the inequality \( f(x) \leq 3 \).

\[ \frac{1 - x + x^2}{1 + x + x^2} \leq 3 \]

Cross-multiplying (assuming \( 1 + x + x^2 > 0 \) for all \( x \in \mathbb{R} \)) gives:

\[ 1 - x + x^2 \leq 3(1 + x + x^2) \]

Expanding the right side:

\[ 1 - x + x^2 \leq 3 + 3x + 3x^2 \]

Rearranging the terms:

\[ 1 - 3 - x - 3x + x^2 - 3x^2 \leq 0 \]

This simplifies to:

\[ -2 - 4x - 2x^2 \leq 0 \]

Dividing through by -2 (which reverses the inequality):

\[ 2 + 2x + x^2 \geq 0 \]

### Step 2: Find the roots of the quadratic

The quadratic \( x^2 + 2x + 2 \) can be analyzed using the discriminant:

\[ D = b^2 - 4ac = 2^2 - 4(1)(2) = 4 - 8 = -4 \]

Since the discriminant is negative, the quadratic has no real roots and is always positive. Thus, \( f(x) \leq 3 \) holds for all \( x \in \mathbb{R} \).

### Step 3: Analyze the function \( g(x) \)

Next, we need to find the maximum value of:

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

### Step 4: Find the critical points of \( g(x) \)

To find the maximum value, we can differentiate \( g(x) \) and set the derivative to zero.

Using the quotient rule:

\[ g'(x) = \frac{(2 + 8x)(1 - 2x + 4x^2) - (1 + 2x + 4x^2)(-2 + 8x)}{(1 - 2x + 4x^2)^2} \]

Setting the numerator equal to zero gives a polynomial equation to solve for \( x \).

### Step 5: Solve for \( x \)

This step involves solving the polynomial equation obtained from the derivative.

### Step 6: Evaluate \( g(x) \) at critical points and endpoints

Once we find the critical points, we evaluate \( g(x) \) at these points and also check the limits as \( x \to \pm\infty \) to find the maximum value.

### Conclusion

After evaluating \( g(x) \) at the critical points, we can determine the maximum value of \( g(x) \).