Textbook Question
Identifying Extrema
In Exercises 53–60:
a. Find the local extrema of each function on the given interval, and say where they occur.
f(x) = sec²x − 2tan x, −π/2 < x < π/2
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5. Graphical Applications of Derivatives
Intro to Extrema
2:06 minutesProblem 4.3.63b
Textbook Question
Identifying Extrema
In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).
b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.
Verified step by step guidance1
Step 1: Observe the graph of f'(x), which represents the derivative of f(x). The derivative indicates the slope of the function f(x). When f'(x) > 0, f(x) is increasing, and when f'(x) < 0, f(x) is decreasing.
Step 2: Identify the intervals where f'(x) is positive (above the x-axis). From the graph, f'(x) is positive on the intervals (-2, -1) and (0, 1). This means f(x) is increasing on these intervals.
Step 3: Identify the intervals where f'(x) is negative (below the x-axis). From the graph, f'(x) is negative on the intervals (-1, 0) and (1, 2). This means f(x) is decreasing on these intervals.
Step 4: Note the points where f'(x) crosses the x-axis. These are the critical points where f'(x) = 0, and f(x) may have local extrema. From the graph, these points occur at x = -1, x = 0, and x = 1.
Step 5: Summarize the behavior of f(x) based on the intervals of positivity and negativity of f'(x). f(x) increases on (-2, -1) and (0, 1), decreases on (-1, 0) and (1, 2), and has potential extrema at x = -1, x = 0, and x = 1.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
First Derivative Test
The First Derivative Test is used to determine where a function is increasing or decreasing. If f'(x) > 0, the function f is increasing, and if f'(x) < 0, f is decreasing. By analyzing the sign changes of f'(x), we can identify intervals where f is positive or negative, which helps in finding local extrema.
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The First Derivative Test: Finding Local Extrema
Critical Points
Critical points occur where the derivative f'(x) is zero or undefined. These points are potential locations for local maxima or minima. In the graph, critical points are where the curve crosses the x-axis, indicating a change in the direction of f(x), which is crucial for identifying extrema.
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Domain of the Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this problem, the domain is (-2, 2), meaning we only consider the behavior of f and f' within this interval. Understanding the domain helps focus the analysis on relevant sections of the graph.
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