Number of points where `f (x)= {{:(max(|x ^(2)- x-2|","x ^(2) -3x),,, x ge0),( max (ln (-x)"," e ^(x)),,, x lt 0):}` is non-differentiable will be:

Options

A. 1
B. 2
C. 3
D. None of these

Answer

To determine the number of points where the function

\[ f(x) = \begin{cases} \max(|x^2 - x - 2|, x^2 - 3x) & \text{for } x \geq 0 \\ \max(\ln(-x), e^x) & \text{for } x < 0 \end{cases} \]

is non-differentiable, we will analyze each case separately.

### Step 1: Analyze for \( x \geq 0 \)

1. **Identify the functions**:
- \( g_1(x) = |x^2 - x - 2| \)
- \( g_2(x) = x^2 - 3x \)

2. **Find the points where \( g_1(x) \) changes**:
- Set \( x^2 - x - 2 = 0 \):
\[ (x - 2)(x + 1) = 0 \implies x = 2 \text{ or } x = -1 \]
- Since we are considering \( x \geq 0 \), we only take \( x = 2 \).

3. **Evaluate \( g_1(x) \) and \( g_2(x) \) at critical points**:
- For \( x = 0 \):
\[ g_1(0) = |0^2 - 0 - 2| = 2, \quad g_2(0) = 0^2 - 3 \cdot 0 = 0 \implies f(0) = \max(2, 0) = 2 \]
- For \( x = 2 \):
\[ g_1(2) = |2^2 - 2 - 2| = |0| = 0, \quad g_2(2) = 2^2 - 3 \cdot 2 = -2 \implies f(2) = \max(0, -2) = 0 \]
- For \( x = 3 \):
\[ g_1(3) = |3^2 - 3 - 2| = |4| = 4, \quad g_2(3) = 3^2 - 3 \cdot 3 = 0 \implies f(3) = \max(4, 0) = 4 \]

4. **Check for non-differentiability**:
- At \( x = 2 \), \( g_1(x) \) changes from \( g_1(x) = -(x^2 - x - 2) \) to \( g_1(x) = x^2 - x - 2 \).
- The derivatives from the left and right at \( x = 2 \) will differ, indicating non-differentiability.

### Step 2: Analyze for \( x < 0 \)

1. **Identify the functions**:
- \( h_1(x) = \ln(-x) \)
- \( h_2(x) = e^x \)

2. **Find the point where \( h_1(x) \) changes**:
- The function \( \ln(-x) \) is defined for \( x < 0 \) and is continuous and differentiable in this interval.
- The function \( e^x \) is also continuous and differentiable for all \( x \).

3. **Check at \( x = -1 \)**:
- At \( x = -1 \):
\[ h_1(-1) = \ln(1) = 0, \quad h_2(-1) = e^{-1} \approx 0.367879 \implies f(-1) = \max(0, 0.367879) = 0.367879 \]
- As \( x \) approaches \( -1 \) from the left, \( \ln(-x) \) approaches \( 0 \) and \( e^x \) approaches \( 0.367879 \).

4. **Check for non-differentiability**:
- At \( x = -1 \), the derivatives of \( h_1(x) \) and \( h_2(x) \) will differ, indicating non-differentiability.

### Conclusion

The function \( f(x) \) is non-differentiable at the following points:
- \( x = 0 \) (discontinuity)
- \( x = 2 \) (sharp point)
- \( x = -1 \) (sharp point)

Thus, the total number of points where \( f(x) \) is non-differentiable is **3**.