Math Problem Statement

give fully explained solution

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|

Solution

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

Function A: f(x)=xsin(x)f(x) = |x| \sin(|x|)

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

Function B: f(x)=xsin(x)f(x) = |x| \sin(\sqrt{|x|})

  • For x>0x > 0, the function becomes f(x)=xsin(x)f(x) = x \sin(\sqrt{x}).
  • For x<0x < 0, it behaves the same since x|x| is used inside the argument. Therefore, we focus on the behavior at x=0x = 0.

To check differentiability at x=0x = 0:

  • As x0x \to 0, sin(x)\sin(\sqrt{x}) behaves as x\sqrt{x}, which tends to 0.
  • The derivative at x=0x = 0 from the right will approach a non-finite value, as the term x\sqrt{|x|} grows too quickly relative to xx.

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

Function C: f(x)=cos(x)f(x) = \cos(|x|)

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

Function D: f(x)=cos(x)f(x) = \cos(\sqrt{|x|})

  • For x>0x > 0, the function behaves as cos(x)\cos(\sqrt{x}), and for x<0x < 0, it is the same due to x|x|.
  • Near x=0x = 0, cos(x)\cos(\sqrt{x}) behaves as 11, but the derivative involves sin(x)12x-\sin(\sqrt{x}) \cdot \frac{1}{2\sqrt{x}}, which tends to infinity as x0x \to 0.

Thus, this function is not differentiable at x=0x = 0 due to the singularity in the derivative caused by the 12x\frac{1}{2\sqrt{x}} term.


Conclusion:

The function not differentiable at x=0x = 0 is f(x)=xsin(x)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 x\sqrt{x} influence differentiability near 0?
  4. Why does cos(x)\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=0x = 0.

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Math Problem Analysis

Mathematical Concepts

Differentiability
Absolute Value
Trigonometric Functions

Formulas

f'(x) = lim(h → 0) (f(x + h) - f(x))/h
Sine and Cosine derivatives: f'(sin x) = cos x, f'(cos x) = -sin x

Theorems

Differentiability at a point requires both continuity and a well-defined derivative from both sides.
Chain Rule for differentiation

Suitable Grade Level

Grades 11-12 (Advanced Calculus)