If `f(x)` satisfies `f(x)+f(3-x)=3 AA x in R`, then the value of integral `I=int_((3)/(4))^((9)/(4))f(x)dx` is equal to

Options

A. 3
B. 6
C. `(9)/(4)`
D. `(9)/(2)`

Answer

To solve the problem, we need to evaluate the integral \( I = \int_{\frac{3}{4}}^{\frac{9}{4}} f(x) \, dx \) given the condition \( f(x) + f(3 - x) = 3 \) for all \( x \in \mathbb{R} \).

### Step-by-Step Solution:

1. **Set up the integral**: We start with the integral we need to evaluate:
\[ I = \int_{\frac{3}{4}}^{\frac{9}{4}} f(x) \, dx \]

2. **Use the property of integrals**: We can use the property of integrals that states:
\[ \int_a^b f(x) \, dx = \int_a^b f(a + b - x) \, dx \]
In our case, \( a = \frac{3}{4} \) and \( b = \frac{9}{4} \). Thus, we can rewrite the integral:
\[ I = \int_{\frac{3}{4}}^{\frac{9}{4}} f(3 - x) \, dx \]

3. **Relate \( f(3 - x) \) to \( f(x) \)**: From the given condition \( f(x) + f(3 - x) = 3 \), we can express \( f(3 - x) \) as:
\[ f(3 - x) = 3 - f(x) \]

4. **Substitute into the integral**: Now we substitute this expression into our integral:
\[ I = \int_{\frac{3}{4}}^{\frac{9}{4}} (3 - f(x)) \, dx \]

5. **Split the integral**: We can split the integral into two parts:
\[ I = \int_{\frac{3}{4}}^{\frac{9}{4}} 3 \, dx - \int_{\frac{3}{4}}^{\frac{9}{4}} f(x) \, dx \]
This simplifies to:
\[ I = 3 \int_{\frac{3}{4}}^{\frac{9}{4}} 1 \, dx - I \]

6. **Evaluate the first integral**: The integral \( \int_{\frac{3}{4}}^{\frac{9}{4}} 1 \, dx \) is simply the length of the interval:
\[ \int_{\frac{3}{4}}^{\frac{9}{4}} 1 \, dx = \frac{9}{4} - \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \]

7. **Substitute back into the equation**: Now substituting back, we have:
\[ I = 3 \cdot \frac{3}{2} - I \]
This simplifies to:
\[ I = \frac{9}{2} - I \]

8. **Solve for \( I \)**: Adding \( I \) to both sides gives:
\[ 2I = \frac{9}{2} \]
Dividing by 2:
\[ I = \frac{9}{4} \]

### Final Answer:
Thus, the value of the integral \( I \) is:
\[ \boxed{\frac{9}{4}} \]