https://file.aiquickdraw.com/mathbot/file/1727580017486u5h07yaf.jpg

The question asks which of the following functions is **not differentiable** at \( x = 0 \). Let's analyze each of the given functions to check their differentiability at \( x = 0 \).

### Function A: \( f(x) = |x| \sin(|x|) \)
- **For \( x > 0 \)**, this function is simply \( f(x) = x \sin(x) \), which is differentiable.
- **For \( x < 0 \)**, the function behaves as \( f(x) = -x \sin(-x) = x \sin(x) \), so the behavior is the same as for positive \( x \).
- At \( x = 0 \), both sides approach 0, and the derivative from the left and right is also continuous. Hence, **differentiable at \( x = 0 \)**.

### Function B: \( f(x) = |x| \sin(\sqrt{|x|}) \)
- For \( x > 0 \), the function becomes \( f(x) = x \sin(\sqrt{x}) \).
- For \( x < 0 \), it behaves the same since \( |x| \) is used inside the argument. Therefore, we focus on the behavior at \( x = 0 \).
  
To check differentiability at \( x = 0 \):
- As \( x \to 0 \), \( \sin(\sqrt{x}) \) behaves as \( \sqrt{x} \), which tends to 0.
- The derivative at \( x = 0 \) from the right will approach a non-finite value, as the term \( \sqrt{|x|} \) grows too quickly relative to \( x \).

Thus, this function **is not differentiable at \( x = 0 \)** due to the sharp behavior caused by \( \sqrt{|x|} \).

### Function C: \( f(x) = \cos(|x|) \)
- This function is continuous and smooth because \( |x| \) does not affect the differentiability of \( \cos(x) \).
- The derivative at \( x = 0 \) is simply \( f'(0) = -\sin(0) = 0 \), so it is **differentiable at \( x = 0 \)**.

### Function D: \( f(x) = \cos(\sqrt{|x|}) \)
- For \( x > 0 \), the function behaves as \( \cos(\sqrt{x}) \), and for \( x < 0 \), it is the same due to \( |x| \).
- Near \( x = 0 \), \( \cos(\sqrt{x}) \) behaves as \( 1 \), but the derivative involves \( -\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}} \), which tends to infinity as \( x \to 0 \).

Thus, this function **is not differentiable at \( x = 0 \)** due to the singularity in the derivative caused by the \( \frac{1}{2\sqrt{x}} \) term.

---

### Conclusion:
The function **not differentiable at \( x = 0 \)** is **\( f(x) = |x| \sin(\sqrt{|x|}) \)** (Option B).

Would you like further details or have any questions? 

Here are five related questions you might find interesting:
1. What are the conditions for differentiability at a point?
2. How does the modulus function affect differentiability?
3. How does \( \sqrt{x} \) influence differentiability near 0?
4. Why does \( \cos(x) \) remain differentiable for all real numbers?
5. How does a function's smoothness relate to its differentiability?

**Tip:** Checking both left-hand and right-hand derivatives is crucial when determining differentiability at sharp points like \( x = 0 \).

Which of the following functions is not differentiable at x = 0? (A) f(x) = |x|sin|x| (B) f(x) = |x|sin√|x| (C) f(x) = cos|x| (D) f(x) = cos√|x|

This problem explores which function is not differentiable at x = 0, using concepts like absolute value and trigonometric functions. A detailed solution examines four functions, their derivatives, and their behavior at x = 0.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation