Math Problem Statement
give fully explained solution
Solution
The question asks which of the following functions is not differentiable at . Let's analyze each of the given functions to check their differentiability at .
Function A:
- For , this function is simply , which is differentiable.
- For , the function behaves as , so the behavior is the same as for positive .
- At , both sides approach 0, and the derivative from the left and right is also continuous. Hence, differentiable at .
Function B:
- For , the function becomes .
- For , it behaves the same since is used inside the argument. Therefore, we focus on the behavior at .
To check differentiability at :
- As , behaves as , which tends to 0.
- The derivative at from the right will approach a non-finite value, as the term grows too quickly relative to .
Thus, this function is not differentiable at due to the sharp behavior caused by .
Function C:
- This function is continuous and smooth because does not affect the differentiability of .
- The derivative at is simply , so it is differentiable at .
Function D:
- For , the function behaves as , and for , it is the same due to .
- Near , behaves as , but the derivative involves , which tends to infinity as .
Thus, this function is not differentiable at due to the singularity in the derivative caused by the term.
Conclusion:
The function not differentiable at is (Option B).
Would you like further details or have any questions?
Here are five related questions you might find interesting:
- What are the conditions for differentiability at a point?
- How does the modulus function affect differentiability?
- How does influence differentiability near 0?
- Why does remain differentiable for all real numbers?
- 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 .
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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)
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