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

The x y-coordinate plane is given. Two curves and a point are on the graph.
The first curve enters the window in the second quadrant, goes horizontally right, at x = −2 goes up and right, crosses the positive y-axis, and stops at the open point in the first quadrant above x = 1.
The closed point is located in the first quadrant directly below the previous open point above x = 1.
The second curve starts at the open point in the first quadrant directly below the previous closed point above x = 1, goes down and right, crosses the x-axis at approximately x = 2.3, goes horizontally right at x = 3, and exits the window in the fourth quadrant.

x=

From the description of the graph, it appears that the function \( f \) has two distinct curves that exhibit features suggesting points where the function is not differentiable. Here’s a breakdown:

1. **At \( x = -2 \)**: The curve makes a sharp turn upwards. Sharp turns or cusps in a graph indicate points where the derivative is undefined because the slope changes abruptly. Therefore, \( f \) is not differentiable at \( x = -2 \).

2. **At \( x = 1 \)**: There is a jump discontinuity between the two curves. The first curve stops at an open point, and the second curve starts from a closed point. A jump discontinuity is a common cause for non-differentiability because the function is not continuous at this point, which is a requirement for differentiability.

3. **At \( x = 3 \)**: The second curve appears to make a sharp horizontal turn at \( x = 3 \). Just like at \( x = -2 \), this sharp turn indicates that the derivative does not exist because the slope abruptly changes.

Thus, the numbers at which \( f \) is not differentiable are:
\[
x = -2, 1, 3
\]

Would you like more details on how differentiability is determined? 

Here are 5 related questions for further practice:
1. What is the relationship between continuity and differentiability?
2. How do you identify points of non-differentiability from a graph?
3. Can a function be continuous at a point but not differentiable?
4. What is the difference between a corner and a cusp in terms of differentiability?
5. How do you determine differentiability analytically for piecewise functions?

**Tip**: For a function to be differentiable at a point, it must be continuous there, and its left-hand and right-hand derivatives must be equal.

The graph of f is given. State the numbers at which f is not differentiable. (Enter your answers as a comma-separated list.) The x y-coordinate plane is given. Two curves and a point are on the graph. The first curve enters the window in the second quadrant, goes horizontally right, at x = −2 goes up and right, crosses the positive y-axis, and stops at the open point in the first quadrant above x = 1. The closed point is located in the first quadrant directly below the previous open point above x = 1. The second curve starts at the open point in the first quadrant directly below the previous closed point above x = 1, goes down and right, crosses the x-axis at approximately x = 2.3, goes horizontally right at x = 3, and exits the window in the fourth quadrant.

Discover how to identify points of non-differentiability from a graph with sharp turns and jump discontinuities. This problem demonstrates key calculus concepts and is suitable for Grades 11-12. Learn about the conditions for differentiability and continuity in graph analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation