If `f(x)+2f(1-x)=6x(AA x in R)`, then the vlaue of `(3)/(4)((f(8))/(f'(1)))` is equal to

Answer

To solve the equation \( f(x) + 2f(1-x) = 6x \) for \( f(x) \) and then find the value of \( \frac{3}{4} \left( \frac{f(8)}{f'(1)} \right) \), we can follow these steps:

### Step 1: Write the original equation
We start with the given equation:
\[ f(x) + 2f(1-x) = 6x \tag{1} \]

### Step 2: Substitute \( x \) with \( 1-x \)
Now, we replace \( x \) with \( 1-x \) in the original equation:
\[ f(1-x) + 2f(x) = 6(1-x) \tag{2} \]

### Step 3: Simplify the second equation
Expanding equation (2):
\[ f(1-x) + 2f(x) = 6 - 6x \]

### Step 4: Set up a system of equations
Now we have two equations:
1. \( f(x) + 2f(1-x) = 6x \) (1)
2. \( f(1-x) + 2f(x) = 6 - 6x \) (2)

### Step 5: Multiply equation (1) by 2
To eliminate \( f(1-x) \), we can multiply equation (1) by 2:
\[ 2f(x) + 4f(1-x) = 12x \tag{3} \]

### Step 6: Subtract equation (2) from equation (3)
Now, we subtract equation (2) from equation (3):
\[ (2f(x) + 4f(1-x)) - (f(1-x) + 2f(x)) = 12x - (6 - 6x) \]
This simplifies to:
\[ 3f(1-x) = 18x - 6 \]
Thus, we can express \( f(1-x) \):
\[ f(1-x) = 6x - 2 \tag{4} \]

### Step 7: Substitute \( f(1-x) \) back into equation (1)
Now, we can substitute equation (4) back into equation (1):
\[ f(x) + 2(6x - 2) = 6x \]
This simplifies to:
\[ f(x) + 12x - 4 = 6x \]
So, we find:
\[ f(x) = 6x - 12x + 4 = 4 - 6x \]

### Step 8: Find the derivative \( f'(x) \)
Now, we differentiate \( f(x) \):
\[ f'(x) = -6 \]

### Step 9: Evaluate \( f'(1) \)
Thus,
\[ f'(1) = -6 \]

### Step 10: Evaluate \( f(8) \)
Now we find \( f(8) \):
\[ f(8) = 4 - 6 \times 8 = 4 - 48 = -44 \]

### Step 11: Calculate \( \frac{3}{4} \left( \frac{f(8)}{f'(1)} \right) \)
Now we can calculate:
\[ \frac{3}{4} \left( \frac{f(8)}{f'(1)} \right) = \frac{3}{4} \left( \frac{-44}{-6} \right) = \frac{3}{4} \times \frac{44}{6} \]
This simplifies to:
\[ = \frac{3 \times 44}{4 \times 6} = \frac{132}{24} = \frac{11}{2} \]

### Final Answer
Thus, the final value is:
\[ \frac{11}{2} = 5.5 \]