If `y= f (x)` is differentiable `AA x in R,` then

Options

A. `y= |f (x)|` is differentiable `AA x in R`
B. `y=f ^(2)(x)` is not-differentiable for atleast one x
C. `y=f (x)|f(x)|` is non-differentiable for atleast one x
D. `y=|f(x)|^(3)` is differentiable `A x in R`

Answer

To solve the problem, we need to analyze the differentiability of the given functions based on the information provided. We know that \( y = f(x) \) is differentiable for all \( x \in \mathbb{R} \). We will evaluate each option one by one.

### Step-by-Step Solution:

1. **Option A: \( y = |f(x)| \)**
- Since \( f(x) \) is differentiable everywhere, we can consider the case when \( f(x) \) crosses zero.
- The function \( |f(x)| \) will have a sharp edge at points where \( f(x) = 0 \).
- Therefore, \( |f(x)| \) is not differentiable at those points.
- **Conclusion**: Option A is incorrect.

2. **Option B: \( y = f^2(x) \)**
- The function \( f^2(x) \) is a composition of differentiable functions.
- The derivative can be computed using the chain rule: \( \frac{dy}{dx} = 2f(x) \cdot f'(x) \).
- Since both \( f(x) \) and \( f'(x) \) are defined for all \( x \in \mathbb{R} \), \( f^2(x) \) is differentiable everywhere.
- **Conclusion**: Option B is correct.

3. **Option C: \( y = f(x) \cdot |f(x)| \)**
- For \( f(x) \cdot |f(x)| \), we analyze the cases:
- When \( f(x) > 0 \), \( |f(x)| = f(x) \) so \( y = f(x) \cdot f(x) = f^2(x) \).
- When \( f(x) < 0 \), \( |f(x)| = -f(x) \) so \( y = f(x) \cdot (-f(x)) = -f^2(x) \).
- At points where \( f(x) = 0 \), we have a transition from \( f^2(x) \) to \( -f^2(x) \), which creates a sharp edge.
- Therefore, it is not differentiable at those points.
- **Conclusion**: Option C is incorrect.

4. **Option D: \( y = |f(x)| \)**
- Similar to Option A, since \( f(x) \) is differentiable, \( |f(x)| \) will have sharp edges at points where \( f(x) = 0 \).
- Thus, it is not differentiable at those points.
- **Conclusion**: Option D is incorrect.

### Final Conclusion:
- The only correct option is **Option B: \( y = f^2(x) \)**.