Let f (x) be a quadratic function such that f (0) =1 and `f(-1)=4, ifint(f(x))/(x^(2)(1+x)^(2))dx` is a rational function then the value of f(10)

Options

A. 584
B. 521
C. 520
D. 583

Explanation

To solve the problem, we will follow these steps:

1. **Define the quadratic function**: Since \( f(x) \) is a quadratic function, we can express it in the general form:
\[ f(x) = ax^2 + bx + c \]

2. **Use the given conditions**: We know that:
- \( f(0) = 1 \)
- \( f(-1) = 4 \)

Using \( f(0) = 1 \):
\[ f(0) = a(0)^2 + b(0) + c = c = 1 \]
So, \( c = 1 \).

3. **Substituting the second condition**: Now, we substitute \( x = -1 \) into the function:
\[ f(-1) = a(-1)^2 + b(-1) + c = a - b + 1 = 4 \]
Substituting \( c = 1 \) into the equation gives:
\[ a - b + 1 = 4 \implies a - b = 3 \quad \text{(Equation 1)} \]

4. **Analyzing the integral condition**: We need to ensure that the integral
\[ \int \frac{f(x)}{x^2 (1+x)^2} \, dx \]
is a rational function. The integral will be rational if the degree of the numerator \( f(x) \) is less than or equal to the degree of the denominator. The denominator \( x^2(1+x)^2 \) has a degree of 4. Since \( f(x) \) is quadratic (degree 2), we need to ensure that the numerator does not introduce any additional poles.

5. **Finding the values of \( a \) and \( b \)**: From Equation 1, we can express \( a \) in terms of \( b \):
\[ a = b + 3 \]

Since \( f(x) \) must not have any poles that would make the integral undefined, we can also check the condition for \( b \). We can assume \( b \) must be such that it does not create any additional singularities in the integral.

6. **Choosing a value for \( b \)**: Let's choose \( b = 1 \) (as it satisfies the condition):
\[ a = 1 + 3 = 4 \]

7. **Final form of the function**: Thus, we have:
\[ f(x) = 4x^2 + 1x + 1 \]

8. **Calculating \( f(10) \)**:
\[ f(10) = 4(10)^2 + 1(10) + 1 = 4 \cdot 100 + 10 + 1 = 400 + 10 + 1 = 411 \]

Thus, the value of \( f(10) \) is \( \boxed{411} \).