Math Problem Statement
The graph of f is given. State the numbers at which f is not differentiable.
Solution
To determine where the function is not differentiable based on the graph, we need to identify points where the function has:
- Sharp corners or cusps: The derivative doesn't exist at these points because the slopes from the left and right are different.
- Discontinuities: If the function jumps or has a break, it's not differentiable there.
- Vertical tangents: The slope becomes infinite, so the function is not differentiable.
Looking at the graph you provided:
- At , there is a corner.
- At , there is a discontinuity (a hole in the graph).
- At , there is a corner.
Thus, the function is not differentiable at .
Would you like further details on differentiability or any specific questions answered? Here are some follow-up questions:
- What is the difference between a corner and a cusp?
- Why is the function not differentiable at points with discontinuities?
- How can vertical tangents affect differentiability?
- How does the concept of limits relate to differentiability?
- Can a continuous function always be differentiable?
Tip: Differentiability implies continuity, but continuity does not necessarily imply differentiability!
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Math Problem Analysis
Mathematical Concepts
Differentiability
Continuity
Graphical Analysis
Formulas
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Theorems
Differentiability Theorem
Continuity and Differentiability
Suitable Grade Level
Grades 11-12 (Calculus)