If G and L are the greatest and least values of the expression`(2x^(2)-3x+2)/(2x^(2)+3x+2), x epsilonR` respectively.
The least value of `G^(100)+L^(100)` is
(a) `2^(100)` (b) `3^(100)` (c) `7^(100)` (d) none of these

Options

A. `2^(100)`
B. `3^(100)`
C. `7^(100)`
D. none of these

Explanation

To solve the problem, we need to find the greatest and least values of the expression

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

for \( x \in \mathbb{R} \), and then compute the least value of \( G^{100} + L^{100} \), where \( G \) is the greatest value and \( L \) is the least value of the expression.

### Step 1: Set the expression equal to \( y \)

We start by letting

\[ y = \frac{2x^2 - 3x + 2}{2x^2 + 3x + 2} \]

Cross-multiplying gives us:

\[ y(2x^2 + 3x + 2) = 2x^2 - 3x + 2 \]

### Step 2: Rearranging the equation

Rearranging this equation leads to:

\[ 2yx^2 + 3yx + 2y - 2x^2 + 3x - 2 = 0 \]

Combining like terms, we have:

\[ (2y - 2)x^2 + (3y + 3)x + (2y - 2) = 0 \]

### Step 3: Using the discriminant

For this quadratic equation in \( x \) to have real solutions, the discriminant must be non-negative:

\[ D = (3y + 3)^2 - 4(2y - 2)(2y - 2) \geq 0 \]

Calculating the discriminant:

\[ D = (3y + 3)^2 - 4(2y - 2)^2 \]

Expanding this gives:

\[ D = 9y^2 + 18y + 9 - 4(4y^2 - 8y + 4) \]
\[ = 9y^2 + 18y + 9 - 16y^2 + 32y - 16 \]
\[ = -7y^2 + 50y - 7 \geq 0 \]

### Step 4: Finding the roots of the quadratic

To find the values of \( y \), we need to solve the quadratic equation:

\[ -7y^2 + 50y - 7 = 0 \]

Using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

\[ y = \frac{-50 \pm \sqrt{50^2 - 4(-7)(-7)}}{2(-7)} \]
\[ = \frac{-50 \pm \sqrt{2500 - 196}}{-14} \]
\[ = \frac{-50 \pm \sqrt{2304}}{-14} \]
\[ = \frac{-50 \pm 48}{-14} \]

Calculating the two roots:

1. \( y = \frac{-2}{-14} = \frac{1}{7} \)
2. \( y = \frac{-98}{-14} = 7 \)

### Step 5: Identifying \( G \) and \( L \)

From the roots, we identify:

- The least value \( L = \frac{1}{7} \)
- The greatest value \( G = 7 \)

### Step 6: Calculating \( G^{100} + L^{100} \)

Now we need to find \( G^{100} + L^{100} \):

\[ G^{100} + L^{100} = 7^{100} + \left(\frac{1}{7}\right)^{100} = 7^{100} + \frac{1}{7^{100}} \]

### Step 7: Finding the least value

The term \( \frac{1}{7^{100}} \) is very small compared to \( 7^{100} \), so the least value of \( G^{100} + L^{100} \) is approximately:

\[ 7^{100} \]

Thus, the least value of \( G^{100} + L^{100} \) is:

\[ \boxed{7^{100}} \]