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The graph of (x) is shown below. State the points where the function f(x) is not differentiable. Explain why the function is not differentiable at each of those points.

The graph of (x) is shown below. State the points where the function f(x) is not differentiable. Explain why the function is not differentiable at each of those points.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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### Question: The graph of \( f(x) \) is shown below. State the points where the function \( f(x) \) is not differentiable. Explain why the function is not differentiable at each of those points. ### Graph Description: The graph presents the function \( f(x) \) displayed on a coordinate plane. The x-axis ranges from -3 to 5, while the y-axis scales vertically with no specific numerical labels except around the origin. In examining the graph: - The function begins from the left, extending downward in a linear fashion until it reaches a minimum point around \( x = -2 \). - At \( x = -2 \), there is a sharp corner indicating a cusp, after which the function changes direction and moves upwards linearly. - The graph then transitions smoothly and continuously, forming a curved shape down toward another minimum point. - At around \( x = 4 \), there is a noticeable discontinuity with an open circle followed by a continuation of the function’s curve that moves upward again. ### Observations: The function is not differentiable at the following points: 1. **x = -2**: At this point, there is a cusp where the function suddenly changes direction, resulting in a sharp corner. The derivative here does not exist because the slopes of the graph from the left and right of \( x = -2 \) do not match and form a sharp angle. 2. **x = 1**: At this point, there is another noticeable sharp corner where the function again changes direction abruptly. The derivative at \( x = 1 \) is undefined due to the abrupt change in the slope of the function. 3. **x = 4**: At this location, there is a clear discontinuity indicated by an open circle. The function does not have a defined value at this point, making it non-differentiable because continuity is a prerequisite for differentiability. Each of these points where the function is not differentiable involves either an abrupt change in direction (a cusp) or a discontinuity (a gap or jump in the function's value), both of which violate the conditions required for differentiability.
Transcribed Image Text:### Question: The graph of \( f(x) \) is shown below. State the points where the function \( f(x) \) is not differentiable. Explain why the function is not differentiable at each of those points. ### Graph Description: The graph presents the function \( f(x) \) displayed on a coordinate plane. The x-axis ranges from -3 to 5, while the y-axis scales vertically with no specific numerical labels except around the origin. In examining the graph: - The function begins from the left, extending downward in a linear fashion until it reaches a minimum point around \( x = -2 \). - At \( x = -2 \), there is a sharp corner indicating a cusp, after which the function changes direction and moves upwards linearly. - The graph then transitions smoothly and continuously, forming a curved shape down toward another minimum point. - At around \( x = 4 \), there is a noticeable discontinuity with an open circle followed by a continuation of the function’s curve that moves upward again. ### Observations: The function is not differentiable at the following points: 1. **x = -2**: At this point, there is a cusp where the function suddenly changes direction, resulting in a sharp corner. The derivative here does not exist because the slopes of the graph from the left and right of \( x = -2 \) do not match and form a sharp angle. 2. **x = 1**: At this point, there is another noticeable sharp corner where the function again changes direction abruptly. The derivative at \( x = 1 \) is undefined due to the abrupt change in the slope of the function. 3. **x = 4**: At this location, there is a clear discontinuity indicated by an open circle. The function does not have a defined value at this point, making it non-differentiable because continuity is a prerequisite for differentiability. Each of these points where the function is not differentiable involves either an abrupt change in direction (a cusp) or a discontinuity (a gap or jump in the function's value), both of which violate the conditions required for differentiability.
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