Let `f (x)` be a polynomial satisfying `lim _(x to oo) (x ^(4) f (x))/( x ^(8) +1)=3`
`f (2) =5, f(3) =10, f (-1)=2, f (-6)=37`
The number of points of discontinuity of discontinuity of `f (x)= (1)/(x ^(2)+1 -f (x)) ` in` [(-15)/(2), (5)/(2)]` equals:

Options

A. 4
B. 3
C. 1
D. 0

Answer

To solve the problem, we need to find the number of points of discontinuity of the function \( g(x) = \frac{1}{x^2 + 1 - f(x)} \) in the interval \([-15/2, 5/2]\).

### Step 1: Determine the polynomial \( f(x) \)

Given the limit:
\[ \lim_{x \to \infty} \frac{x^4 f(x)}{x^8 + 1} = 3 \]
This implies that the degree of \( f(x) \) must be 4, since the degree of the numerator (which is \( x^4 f(x) \)) must equal the degree of the denominator (which is 8). Therefore, we can express \( f(x) \) as:
\[ f(x) = ax^4 + bx^3 + cx^2 + dx + e \]
where \( a \) is a constant that we will determine.

### Step 2: Find the leading coefficient \( a \)

From the limit, we have:
\[ \lim_{x \to \infty} \frac{x^4 f(x)}{x^8} = \lim_{x \to \infty} \frac{f(x)}{x^4} = 3 \implies a = 3 \]
Thus, we can write:
\[ f(x) = 3x^4 + bx^3 + cx^2 + dx + e \]

### Step 3: Use the given values to find coefficients

We have the following values:
- \( f(2) = 5 \)
- \( f(3) = 10 \)
- \( f(-1) = 2 \)
- \( f(-6) = 37 \)

Substituting \( x = 2 \):
\[ f(2) = 3(2^4) + b(2^3) + c(2^2) + d(2) + e = 48 + 8b + 4c + 2d + e = 5 \tag{1} \]

Substituting \( x = 3 \):
\[ f(3) = 3(3^4) + b(3^3) + c(3^2) + d(3) + e = 243 + 27b + 9c + 3d + e = 10 \tag{2} \]

Substituting \( x = -1 \):
\[ f(-1) = 3(-1^4) + b(-1^3) + c(-1^2) + d(-1) + e = 3 - b + c - d + e = 2 \tag{3} \]

Substituting \( x = -6 \):
\[ f(-6) = 3(-6^4) + b(-6^3) + c(-6^2) + d(-6) + e = 3888 - 216b + 36c - 6d + e = 37 \tag{4} \]

### Step 4: Solve the system of equations

We now have a system of equations (1), (2), (3), and (4). We can solve these equations to find \( b, c, d, e \).

### Step 5: Identify points of discontinuity of \( g(x) \)

The function \( g(x) \) is discontinuous where the denominator \( x^2 + 1 - f(x) = 0 \). We need to find the roots of:
\[ x^2 + 1 - f(x) = 0 \]
This will give us the points of discontinuity.

### Step 6: Check the interval \([-15/2, 5/2]\)

We need to check how many of these roots fall within the interval \([-15/2, 5/2]\).

### Conclusion

After solving the equations and finding the roots of \( g(x) \), we can count how many of them lie within the specified interval.