Let `f:[-1/2,2] rarr R` and `g:[-1/2,2] rarr R` be functions defined by `f(x)=[x^2-3]` and `g(x)=|x|f(x)+|4x-7|f(x)`, where [y] denotes the greatest integer less than or equal to y for `yinR`. Then,

Options

A. f is discontinuous exactly at three points in [-1/2,2]
B. f is discontinuous exactly at four points in [-1/2,2]
C. g is not differentiable exactly at four points in [-1/2,2]
D. g is not differentiable exactly at five points in [-1/2,2]

Explanation

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) defined in the question.

1. **Define the functions**:
- The function \( f(x) \) is given by:
\[ f(x) = x^2 - 3 \]
- The function \( g(x) \) is defined as:
\[ g(x) = |x| f(x) + |4x - 7| f(x) \]

2. **Analyze \( f(x) \)**:
- The function \( f(x) = x^2 - 3 \) is a quadratic function. We need to determine its continuity and differentiability.
- The range of \( f(x) \) when \( x \) is in the interval \([-1/2, 2]\):
- At \( x = -1/2 \):
\[ f(-1/2) = \left(-\frac{1}{2}\right)^2 - 3 = \frac{1}{4} - 3 = -\frac{11}{4} \]
- At \( x = 2 \):
\[ f(2) = 2^2 - 3 = 4 - 3 = 1 \]
- Therefore, \( f(x) \) takes values from \(-\frac{11}{4}\) to \(1\). Since \( f(x) \) is a polynomial, it is continuous and differentiable everywhere in its domain.

3. **Analyze \( g(x) \)**:
- The function \( g(x) \) can be rewritten as:
\[ g(x) = |x| (x^2 - 3) + |4x - 7| (x^2 - 3) \]
- We need to analyze \( g(x) \) at critical points where the absolute values change, specifically at \( x = 0 \) and \( x = \frac{7}{4} \).

4. **Evaluate \( g(x) \) at critical points**:
- For \( x < 0 \):
\[ g(x) = -x (x^2 - 3) + (7 - 4x)(x^2 - 3) \]
- For \( 0 \leq x < \frac{7}{4} \):
\[ g(x) = x (x^2 - 3) + (7 - 4x)(x^2 - 3) \]
- For \( x \geq \frac{7}{4} \):
\[ g(x) = x (x^2 - 3) + (4x - 7)(x^2 - 3) \]

5. **Check continuity and differentiability**:
- At \( x = 0 \):
- Left-hand limit: \( g(0^-) \)
- Right-hand limit: \( g(0^+) \)
- At \( x = \frac{7}{4} \):
- Left-hand limit: \( g\left(\frac{7}{4}^-\right) \)
- Right-hand limit: \( g\left(\frac{7}{4}^+\right) \)

6. **Conclusion**:
- \( f(x) \) is continuous and differentiable everywhere in \([-1/2, 2]\).
- \( g(x) \) is not differentiable at \( x = 0 \) and \( x = \frac{7}{4} \) due to the absolute value functions.

### Summary of Results:
- \( f(x) \) is continuous and differentiable on \([-1/2, 2]\).
- \( g(x) \) is not differentiable at \( x = 0 \) and \( x = \frac{7}{4} \).