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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.63b
Chapter 4, Problem 4.3.63b

Identifying Extrema


In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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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.

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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.

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Key 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.
Recommended video:
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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.
Recommended video:
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Critical Points

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.
Recommended video:
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Finding the Domain and Range of a Graph
Related Practice
Textbook Question

114. Parabolas

b. When is the parabola concave up? Concave down?

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Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

-sec²(3x/2)

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Textbook Question

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


b. Does f'(3) exist?

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Textbook Question

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local maximum at (1, 1) and a local minimum at (3, 3).

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Textbook Question

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:



b. On what open intervals is f increasing or decreasing?


f′(x) = (x − 1)(x + 2)(x − 3)

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Textbook Question

Finding displacement from an antiderivative of velocity

a. Suppose that the velocity of a body moving along the s-axis is

ds/dt = v = 9.8t − 3.

iii. Now find the body’s displacement from t = 1 to t = 3 given that s = s₀ when t = 0.

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