Textbook Question
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = -x⁴ - 2x³ + 12x²
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5. Graphical Applications of Derivatives
Concavity
2:26 minutesProblem 107c
Textbook Question
The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>
c. At what point(s) does f have an inflection point?
Verified step by step guidance1
Understand that an inflection point occurs where the second derivative changes sign, indicating a change in concavity of the function f.
Since the graph of f' is provided, we need to analyze the behavior of f' to infer information about f''.
Identify where f' has local extrema (maximum or minimum points) on the interval [-3, 2]. These points are where f'' changes sign.
Look for points on the graph of f' where it changes from increasing to decreasing or vice versa. These are potential inflection points for f.
Verify that at these points, the sign of f'' changes by considering the slope of f' before and after these points.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First Derivative and Critical Points
The first derivative of a function, denoted as f', indicates the rate of change of the function f. Critical points occur where f' is zero or undefined, which can signify local maxima, minima, or points of inflection. Understanding these points is essential for analyzing the behavior of the function and determining where it changes direction.
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Critical Points
Second Derivative and Inflection Points
An inflection point occurs where the concavity of the function changes, which is determined by the second derivative, f''. If f'' changes sign at a point, it indicates a transition in concavity, suggesting that the function f has an inflection point there. Identifying these points is crucial for understanding the overall shape of the graph.
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Graphical Interpretation of Derivatives
The graph of the first derivative f' provides valuable insights into the behavior of the original function f. By analyzing where f' is positive or negative, one can infer where f is increasing or decreasing. Additionally, the points where f' crosses the x-axis indicate potential critical points, which are essential for locating inflection points in the context of the original function.
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