If f(x) satisfies the condition for Rolle's heorem on [3,5] then `int_(3)^(5) f(x)` dx equals

Options

A. 2
B. `-1`
C. 0
D. `-4//3`

Explanation

To solve the problem, we need to find the value of the integral \( \int_{3}^{5} f(x) \, dx \) given that \( f(x) \) satisfies the conditions of Rolle's theorem on the interval \([3, 5]\).

### Step-by-Step Solution:

1. **Understanding Rolle's Theorem**:
Since \( f(x) \) satisfies the conditions of Rolle's theorem, it must be continuous on the closed interval \([3, 5]\) and differentiable on the open interval \((3, 5)\). Additionally, \( f(3) = f(5) \).

2. **Setting Up the Function**:
From the conditions of Rolle's theorem, we know that \( f(3) = f(5) = 0 \). This implies that \( 3 \) and \( 5 \) are roots of the function \( f(x) \). Therefore, we can express \( f(x) \) in the form:
\[ f(x) = k(x - 3)(x - 5) \]
where \( k \) is a constant.

3. **Choosing a Simple Form for \( f(x) \)**:
For simplicity, we can choose \( k = 1 \) (the specific value of \( k \) does not affect the integral since it will be multiplied by the same constant). Thus:
\[ f(x) = (x - 3)(x - 5) = x^2 - 8x + 15 \]

4. **Integrating \( f(x) \)**:
Now we need to compute the integral:
\[ \int_{3}^{5} f(x) \, dx = \int_{3}^{5} (x^2 - 8x + 15) \, dx \]

5. **Calculating the Integral**:
We will integrate term by term:
\[ \int (x^2 - 8x + 15) \, dx = \frac{x^3}{3} - 4x^2 + 15x \]
Now we evaluate this from \( 3 \) to \( 5 \):
\[ \left[ \frac{x^3}{3} - 4x^2 + 15x \right]_{3}^{5} \]

6. **Evaluating at the Limits**:
First, evaluate at \( x = 5 \):
\[ \frac{5^3}{3} - 4(5^2) + 15(5) = \frac{125}{3} - 100 + 75 = \frac{125}{3} - 25 = \frac{125 - 75}{3} = \frac{50}{3} \]
Now, evaluate at \( x = 3 \):
\[ \frac{3^3}{3} - 4(3^2) + 15(3) = \frac{27}{3} - 36 + 45 = 9 - 36 + 45 = 18 \]

7. **Finding the Result**:
Now, subtract the two results:
\[ \int_{3}^{5} f(x) \, dx = \left( \frac{50}{3} \right) - 18 \]
Convert \( 18 \) to a fraction:
\[ 18 = \frac{54}{3} \]
Thus:
\[ \int_{3}^{5} f(x) \, dx = \frac{50}{3} - \frac{54}{3} = \frac{50 - 54}{3} = \frac{-4}{3} \]

### Final Answer:
\[ \int_{3}^{5} f(x) \, dx = -\frac{4}{3} \]