Textbook Question
Theory and Examples
Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).
5. Graphical Applications of Derivatives
Intro to Extrema
4:25 minutesProblem 4.3.15
Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.
Verified step by step guidance1
Step 1: Analyze the graph to determine where the function is increasing or decreasing. Look for intervals where the slope of the graph is positive (increasing) or negative (decreasing).
Step 2: Identify the intervals of increase. From the graph, observe that the function is increasing on the intervals (-3, -2) and (0, 1).
Step 3: Identify the intervals of decrease. From the graph, observe that the function is decreasing on the intervals (-2, 0) and (1, 3).
Step 4: Determine the local extrema by finding the points where the function changes from increasing to decreasing or vice versa. These are typically at the peaks and troughs of the graph.
Step 5: Identify the local and absolute extrema. From the graph, the local maximum occurs at x = -2 and x = 1, and the local minimum occurs at x = 0 and x = 3. The absolute minimum is at x = 3, and the absolute maximum is at x = -2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Increasing and Decreasing Functions
A function is said to be increasing on an interval if, for any two points within that interval, the function's value at the second point is greater than at the first. Conversely, a function is decreasing if the value at the second point is less than at the first. Identifying these intervals involves analyzing the function's derivative, where positive values indicate increasing behavior and negative values indicate decreasing behavior.
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Local and Absolute Extrema
Local extrema refer to points where a function reaches a maximum or minimum value within a specific neighborhood, while absolute extrema are the highest or lowest values of the function over its entire domain. To find these extrema, one typically uses the first and second derivative tests, evaluating critical points where the derivative is zero or undefined, and determining their nature through further analysis.
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Critical Points
Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local extrema, as they represent potential locations where the function changes from increasing to decreasing or vice versa. Analyzing critical points helps in determining the behavior of the function and locating its maxima and minima.
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