Math Problem Statement

The graph of f is given. State the numbers at which f is not differentiable. (Enter your answers as a comma-separated list.)

Solution

I can see that you've uploaded a graph of the function f(x)f(x), and you're asked to determine where ff is not differentiable. To analyze this:

  1. The function is not differentiable at points where there are sharp corners or cusp points, as these represent places where the slope changes abruptly.
  2. It is also not differentiable at points where there are vertical tangents or discontinuities in the graph.

Let's analyze the graph you provided and identify the points where the function is not differentiable. I'll carefully examine the sharp turns and any other potential issues with differentiability.

Give me a moment to review it. PIL import Image

Load the image to analyze its content

image_path = '/mnt/data/1741065145296hnglc22i.jpg' img = Image.open(image_path) img.show()I have displayed the image for analysis. Based on the graph's features, the function f(x)f(x) appears to have some specific characteristics that we need to focus on to determine where it's not differentiable.

From the graph:

  1. At x=2x = -2, there is a sharp corner where the graph changes direction.
  2. At x=2x = 2, there appears to be another sharp turn or cusp.

Thus, the function is not differentiable at x=2x = -2 and x=2x = 2. These points are where the graph has sharp corners or cusps, indicating non-differentiability.

So, the answer is:
2,2-2, 2

Would you like further explanation on why these points are not differentiable? Or do you have any other questions related to this problem?

Here are a few related questions to explore:

  1. How do you determine differentiability at a point using the derivative?
  2. What is the difference between continuity and differentiability at a point?
  3. How do you handle piecewise functions when analyzing differentiability?
  4. What are the common types of discontinuities in graphs?
  5. Can a function be continuous but not differentiable?

Tip: Always check for sharp corners, cusps, vertical tangents, and discontinuities when analyzing differentiability.

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Math Problem Analysis

Mathematical Concepts

Differentiability
Calculus
Graph Analysis

Formulas

-

Theorems

Differentiability Criterion

Suitable Grade Level

Grades 11-12