Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = cos² x on [-π,π]
5. Graphical Applications of Derivatives
Intro to Extrema
2:28 minutesProblem 4.R.2b
Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
b. Give the approximate coordinates of the absolute maximum and minimum values of ƒ (if they exist).
Verified step by step guidance1
Begin by understanding the concept of extrema. Extrema refer to the maximum and minimum values of a function within a given interval. Absolute extrema are the highest and lowest points over the entire interval.
Examine the graph of the function ƒ on the interval [-3, 3]. Look for the highest point on the graph within this interval to identify the absolute maximum, and the lowest point to identify the absolute minimum.
Check the endpoints of the interval, x = -3 and x = 3, as extrema can occur at these points. Compare the function values at these endpoints with other points within the interval.
Identify any critical points within the interval [-3, 3]. Critical points occur where the derivative of the function is zero or undefined. These points can potentially be locations of extrema.
After identifying potential extrema points, compare their function values to determine which is the absolute maximum and which is the absolute minimum. Remember, the absolute maximum is the largest value, and the absolute minimum is the smallest value within the interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Extrema
Extrema refer to the maximum and minimum values of a function within a given interval. An absolute maximum is the highest point on the graph over that interval, while an absolute minimum is the lowest point. Identifying these points is crucial for understanding the behavior of the function and can involve evaluating the function at critical points and endpoints.
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Critical Points
Critical points are values in the domain of a function where its derivative is either zero or undefined. These points are essential for locating extrema, as they indicate where the function may change direction. To find absolute extrema, one must evaluate the function at these critical points as well as at the endpoints of the interval.
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Closed Interval
A closed interval, denoted as [a, b], includes its endpoints a and b. In calculus, analyzing functions over closed intervals is important because it guarantees that the function will attain both maximum and minimum values within that range. This is a key aspect of the Extreme Value Theorem, which states that continuous functions on closed intervals achieve their extrema.
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