Textbook Question
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
h(t) = 2 + cos 2t on [0,π]
5. Graphical Applications of Derivatives
Concavity
2:28 minutesProblem 4.R.5b
Textbook Question
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.
Verified step by step guidance1
First, understand that inflection points occur where the second derivative, ƒ'', changes sign. This means you need to analyze the graph of ƒ'' to identify where it crosses the x-axis.
Look at the graph of ƒ'' and identify the x-values where the graph crosses the x-axis. These x-values are potential inflection points because they indicate a change in concavity.
To determine the intervals of concavity, observe the sign of ƒ'' on either side of each inflection point. If ƒ'' is positive, the function ƒ is concave up; if ƒ'' is negative, the function ƒ is concave down.
List the intervals where ƒ'' is positive and label them as concave up intervals. Similarly, list the intervals where ƒ'' is negative and label them as concave down intervals.
Verify the inflection points by ensuring that the sign of ƒ'' changes at these points. This confirms that the function ƒ changes concavity at these x-values.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inflection Points
Inflection points are points on the graph of a function where the concavity changes. This occurs when the second derivative of the function, f'', is equal to zero or undefined. Identifying these points is crucial for understanding the behavior of the function, as they indicate where the graph shifts from being concave up to concave down, or vice versa.
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Critical Points
Concavity
Concavity refers to the direction in which a curve bends. A function is concave up on an interval if its second derivative, f'', is positive, indicating that the slope of the tangent line is increasing. Conversely, a function is concave down if f'' is negative, meaning the slope of the tangent line is decreasing. Understanding concavity helps in analyzing the overall shape of the graph.
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Second Derivative Test
The second derivative test is a method used to determine the concavity of a function and locate inflection points. By examining the sign of the second derivative, f'', one can ascertain whether the function is concave up or down. If f'' changes sign at a point, that point is an inflection point, providing valuable information about the function's behavior in that vicinity.
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