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 problem, we start with the given equation:

\[ f(x) + 2f(1-x) = 6x \]

### Step 1: Substitute \( x \) with \( 1-x \)

We replace \( x \) with \( 1-x \) in the original equation:

\[ f(1-x) + 2f(x) = 6(1-x) \]

This simplifies to:

\[ f(1-x) + 2f(x) = 6 - 6x \]

### Step 2: Rearranging the equations

Now we have two equations:

1. \( f(x) + 2f(1-x) = 6x \) (Equation 1)
2. \( f(1-x) + 2f(x) = 6 - 6x \) (Equation 2)

### Step 3: Multiply Equation 2 by 2

To eliminate \( f(1-x) \), we multiply Equation 2 by 2:

\[ 2f(1-x) + 4f(x) = 12 - 12x \]

### Step 4: Subtract Equation 1 from the modified Equation 2

Now we subtract Equation 1 from the modified Equation 2:

\[ (2f(1-x) + 4f(x)) - (f(x) + 2f(1-x)) = (12 - 12x) - (6x) \]

This simplifies to:

\[ 3f(x) = 12 - 18x \]

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

Now we can solve for \( f(x) \):

\[ f(x) = \frac{12 - 18x}{3} = 4 - 6x \]

### Step 6: Find \( f'(x) \)

Now we differentiate \( f(x) \):

\[ f'(x) = -6 \]

### Step 7: Calculate \( f(8) \)

Next, we find \( f(8) \):

\[ f(8) = 4 - 6(8) = 4 - 48 = -44 \]

### Step 8: Calculate \( f'(1) \)

Since \( f'(x) = -6 \), we have:

\[ f'(1) = -6 \]

### Step 9: Calculate \( \frac{3}{4} \left( \frac{f(8)}{f'(1)} \right) \)

Now we can calculate the final expression:

\[ \frac{3}{4} \left( \frac{f(8)}{f'(1)} \right) = \frac{3}{4} \left( \frac{-44}{-6} \right) = \frac{3}{4} \cdot \frac{44}{6} = \frac{3 \cdot 44}{4 \cdot 6} = \frac{132}{24} = \frac{11}{2} = 5.5 \]

### Final Answer

Thus, the value of \( \frac{3}{4} \left( \frac{f(8)}{f'(1)} \right) \) is:

\[ \boxed{5.5} \]