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 \( f(x) \) for \( x \geq 0 \)

For \( x \geq 0 \), we need to consider the two functions \( |x^2 - x - 2| \) and \( x^2 - 3x \).

1. **Find the roots of \( x^2 - x - 2 \)**:
\[ x^2 - x - 2 = 0 \implies (x - 2)(x + 1) = 0 \implies x = 2, -1 \]
Since we are only interested in \( x \geq 0 \), we take \( x = 2 \).

2. **Evaluate \( |x^2 - x - 2| \)**:
- For \( 0 \leq x < 2 \), \( x^2 - x - 2 < 0 \) so \( |x^2 - x - 2| = -(x^2 - x - 2) = -x^2 + x + 2 \).
- For \( x = 2 \), \( |x^2 - x - 2| = 0 \).
- For \( x > 2 \), \( x^2 - x - 2 \geq 0 \) so \( |x^2 - x - 2| = x^2 - x - 2 \).

3. **Find the roots of \( x^2 - 3x \)**:
\[ x^2 - 3x = 0 \implies x(x - 3) = 0 \implies x = 0, 3 \]

### Step 2: Determine the intervals for \( f(x) \)

- For \( 0 \leq x < 2 \):
\[ f(x) = \max(-x^2 + x + 2, x^2 - 3x) \]
Evaluate at \( x = 0 \):
\[ f(0) = \max(2, 0) = 2 \]
Evaluate at \( x = 1 \):
\[ f(1) = \max(2, -2) = 2 \]
Evaluate at \( x = 2 \):
\[ f(2) = \max(0, -2) = 0 \]

- For \( 2 < x < 3 \):
\[ f(x) = \max(x^2 - x - 2, x^2 - 3x) = x^2 - 3x \]

- For \( x \geq 3 \):
\[ f(x) = x^2 - x - 2 \]

### Step 3: Points of non-differentiability for \( x \geq 0 \)

1. **At \( x = 0 \)**:
- \( f(x) \) changes from \( 2 \) to \( 0 \) (discontinuity).

2. **At \( x = 2 \)**:
- The function transitions from \( -x^2 + x + 2 \) to \( x^2 - 3x \) (abrupt change).

3. **At \( x = 3 \)**:
- The function transitions from \( x^2 - 3x \) to \( x^2 - x - 2 \) (abrupt change).

### Step 4: Analyze \( f(x) \) for \( x < 0 \)

For \( x < 0 \):
- \( f(x) = \max(\ln(-x), e^x) \)

1. **Evaluate at \( x = -1 \)**:
\[ f(-1) = \max(\ln(1), e^{-1}) = \max(0, \frac{1}{e}) = \frac{1}{e} \]

2. **Behavior as \( x \to 0^- \)**:
- \( \ln(-x) \to -\infty \) and \( e^x \to 1 \), hence \( f(x) \to 1 \).

### Step 5: Points of non-differentiability for \( x < 0 \)

- At \( x = 0 \): The function is not defined (discontinuity).
- The function \( \ln(-x) \) and \( e^x \) are both continuous, but their maximum function may have a point of non-differentiability depending on their slopes.

### Conclusion

The total points of non-differentiability are:
- \( x = 0 \) (discontinuity)
- \( x = 2 \) (abrupt change)
- \( x = 3 \) (abrupt change)

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

### Final Answer:
The number of points where \( f(x) \) is non-differentiable is **3**.

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