`f(x) = sgn(x^2 - ax + 1)` has maximum number of points of discontinuity then

Options

A. ` a in (2,oo)`
B. `a in (-oo,-2)`
C. `a in (-2,2)`
D. none of these

Answer

To solve the problem, we need to determine the maximum number of points of discontinuity for the function \( f(x) = \text{sgn}(x^2 - ax + 1) \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) \) is defined as the signum function of the quadratic expression \( g(x) = x^2 - ax + 1 \). The signum function, \( \text{sgn}(y) \), is defined as:
- \( \text{sgn}(y) = 1 \) if \( y > 0 \)
- \( \text{sgn}(y) = 0 \) if \( y = 0 \)
- \( \text{sgn}(y) = -1 \) if \( y < 0 \)

2. **Finding Points of Discontinuity**:
The function \( f(x) \) will be discontinuous at points where \( g(x) = 0 \). Therefore, we need to find the roots of the quadratic equation:
\[ x^2 - ax + 1 = 0 \]

3. **Using the Discriminant**:
The roots of the quadratic equation can be found using the discriminant \( D \):
\[ D = b^2 - 4ac = (-a)^2 - 4 \cdot 1 \cdot 1 = a^2 - 4 \]
For the quadratic to have two distinct real roots, the discriminant must be greater than zero:
\[ a^2 - 4 > 0 \]

4. **Solving the Inequality**:
We solve the inequality \( a^2 - 4 > 0 \):
\[ a^2 > 4 \]
This implies:
\[ a < -2 \quad \text{or} \quad a > 2 \]

5. **Conclusion**:
The values of \( a \) that satisfy this inequality will ensure that the quadratic \( g(x) \) has two distinct real roots, leading to two points of discontinuity in \( f(x) \). Hence, the maximum number of points of discontinuity for the function \( f(x) \) is 2.

### Final Answer:
The maximum number of points of discontinuity of the function \( f(x) = \text{sgn}(x^2 - ax + 1) \) occurs when \( a < -2 \) or \( a > 2 \).