Consider the function `f (x) and g (x),` both defined from `R to R`
`f (x) = (x ^(3))/(2 )+1 -x int _(0)^(x) g (t) dt and g (x) =x - int _(0) ^(1) f (t) dt, ` then
The number of points of intersection of `f (x) and g (x)` is/are:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To find the number of points of intersection between the functions \( f(x) \) and \( g(x) \), we start by analyzing the given functions:

1. **Define the Functions**:
- \( f(x) = \frac{x^3}{2} + 1 - x \int_0^x g(t) dt \)
- \( g(x) = x - \int_0^1 f(t) dt \)

2. **Substituting \( g(x) \) into \( f(x) \)**:
- We need to express \( g(x) \) in a more manageable form. Since \( g(x) \) contains a definite integral of \( f(t) \), we denote:
\[ A = \int_0^1 f(t) dt \]
- Thus, we can rewrite \( g(x) \) as:
\[ g(x) = x - A \]

3. **Substituting \( g(t) \) into \( f(x) \)**:
- Now, we substitute \( g(t) = t - A \) into the expression for \( f(x) \):
\[ f(x) = \frac{x^3}{2} + 1 - x \int_0^x (t - A) dt \]

4. **Calculating the Integral**:
- We need to compute the integral \( \int_0^x (t - A) dt \):
\[ \int_0^x (t - A) dt = \int_0^x t dt - \int_0^x A dt = \left[ \frac{t^2}{2} \right]_0^x - A \left[ t \right]_0^x = \frac{x^2}{2} - Ax \]
- Therefore, substituting this back into \( f(x) \):
\[ f(x) = \frac{x^3}{2} + 1 - x \left( \frac{x^2}{2} - Ax \right) = \frac{x^3}{2} + 1 - \left( \frac{x^3}{2} - Ax^2 \right) \]
- Simplifying gives:
\[ f(x) = 1 + Ax^2 \]

5. **Setting \( f(x) \) equal to \( g(x) \)**:
- Now, we set \( f(x) \) equal to \( g(x) \):
\[ 1 + Ax^2 = x - A \]
- Rearranging this equation gives:
\[ Ax^2 - x + (A + 1) = 0 \]

6. **Finding the Discriminant**:
- The number of solutions (points of intersection) depends on the discriminant of the quadratic equation:
\[ D = b^2 - 4ac = (-1)^2 - 4A(A + 1) = 1 - 4A^2 - 4A \]

7. **Analyzing the Discriminant**:
- We need to determine when the discriminant is non-negative:
\[ D = 1 - 4A^2 - 4A \]
- This is a quadratic in \( A \). To find the roots, we can use the quadratic formula:
\[ A = \frac{-4 \pm \sqrt{16 + 16}}{2 \cdot -4} = \frac{-4 \pm 4\sqrt{2}}{-8} = \frac{1 \mp \sqrt{2}}{2} \]

8. **Conclusion**:
- Depending on the value of \( A \), we can determine the number of points of intersection:
- If \( D > 0 \): Two points of intersection.
- If \( D = 0 \): One point of intersection.
- If \( D < 0 \): No points of intersection.
- Since we find that the discriminant can be negative for certain values of \( A \), we conclude that there may be no points of intersection.

Thus, the final answer is that there are **no points of intersection** between \( f(x) \) and \( g(x) \).