Math Problem Statement

The graph of f is given. State the numbers at which f is not differentiable.

Solution

To determine where the function f(x)f(x) is not differentiable based on the graph, we need to identify points where the function has:

  1. Sharp corners or cusps: The derivative doesn't exist at these points because the slopes from the left and right are different.
  2. Discontinuities: If the function jumps or has a break, it's not differentiable there.
  3. Vertical tangents: The slope becomes infinite, so the function is not differentiable.

Looking at the graph you provided:

  1. At x=4x = -4, there is a corner.
  2. At x=0x = 0, there is a discontinuity (a hole in the graph).
  3. At x=2x = 2, there is a corner.

Thus, the function f(x)f(x) is not differentiable at x=4,0,2x = -4, 0, 2.

Would you like further details on differentiability or any specific questions answered? Here are some follow-up questions:

  1. What is the difference between a corner and a cusp?
  2. Why is the function not differentiable at points with discontinuities?
  3. How can vertical tangents affect differentiability?
  4. How does the concept of limits relate to differentiability?
  5. 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)