Which of the following function is non- differentiable ?

Options

A. `f(x) =(e^(x)-1)|e^(2x)-1|"in" R`
B. `f(x) =(x-1)/(x^(2)+1)i n R `
C. `f(x) ={{:(||x-3|-1|, xlt3),((x)/(3)[x] -2 , x ge 3):}`
where [.] represents the greatest integer function
D. `f(x) =3(x-2)^((3)/(4))+3 i n R`

Answer

To determine which of the given functions is non-differentiable, we will analyze each option step by step.

### Step 1: Analyze the first function
The first function is given as:
\[ f(x) = e^x - 1 + |e^{2x} - 1| \]

1. **Break down the absolute value**:
\[ |e^{2x} - 1| = \begin{cases} e^{2x} - 1 & \text{if } e^{2x} \geq 1 \\ -(e^{2x} - 1) & \text{if } e^{2x} < 1 \end{cases} \]
Since \( e^{2x} \) is always positive, it will never be less than 1 for any real \( x \). Thus, we can simplify:
\[ |e^{2x} - 1| = e^{2x} - 1 \]

2. **Combine the terms**:
\[ f(x) = e^x - 1 + (e^{2x} - 1) = e^x + e^{2x} - 2 \]

3. **Check differentiability**:
The function \( e^x \) and \( e^{2x} \) are both differentiable everywhere. Therefore, \( f(x) \) is differentiable.

### Step 2: Analyze the second function
The second function is:
\[ f(x) = \frac{1}{x - 1} \]

1. **Check for points of non-differentiability**:
The function is a rational function and is differentiable everywhere except where the denominator is zero. The denominator \( x - 1 = 0 \) at \( x = 1 \).

2. **Conclusion**:
The function is non-differentiable at \( x = 1 \).

### Step 3: Analyze the third function
The third function is:
\[ f(x) = |x - 3| - |x - 3| - 1 \text{ for } x < 3 \]
\[ f(x) = \frac{x}{3} \text{ for } x \geq 3 \]

1. **Break down the absolute values**:
For \( x < 3 \):
\[ f(x) = 3 - x - 1 = 2 - x \]
For \( x \geq 3 \):
\[ f(x) = \frac{x}{3} \]

2. **Check differentiability at \( x = 3 \)**:
- Left-hand derivative at \( x = 3 \):
\[ f'(x) = -1 \]
- Right-hand derivative at \( x = 3 \):
\[ f'(x) = \frac{1}{3} \]

Since the left-hand and right-hand derivatives at \( x = 3 \) are not equal, the function is non-differentiable at \( x = 3 \).

### Step 4: Analyze the fourth function
The fourth function is:
\[ f(x) = \lfloor x \rfloor - 2 \text{ for } x < 2 \]
\[ f(x) = \frac{3}{4}x \text{ for } x \geq 2 \]

1. **Check for points of non-differentiability**:
The floor function \( \lfloor x \rfloor \) is not differentiable at integer points. At \( x = 2 \), the function changes from a step function to a linear function.

2. **Conclusion**:
The function is non-differentiable at \( x = 2 \).

### Final Conclusion
After analyzing all the functions, we find that:
- The first function is differentiable.
- The second function is non-differentiable at \( x = 1 \).
- The third function is non-differentiable at \( x = 3 \).
- The fourth function is non-differentiable at \( x = 2 \).

**Thus, the functions that are non-differentiable are the second, third, and fourth functions.**