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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 4.3.1c
Chapter 4, Problem 4.3.1c

Analyzing Functions from Derivatives


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


c. At what points, if any, does f assume local maximum or minimum values?


f′(x) = x(x − 1)

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1
To find the points where the function f assumes local maximum or minimum values, we need to identify the critical points. Critical points occur where the derivative f'(x) is zero or undefined. Start by setting the derivative equal to zero: f'(x) = x(x - 1) = 0.
Solve the equation x(x - 1) = 0 to find the critical points. This equation can be factored into two parts: x = 0 and x - 1 = 0. Solving these gives the critical points x = 0 and x = 1.
Next, determine whether each critical point is a local maximum, minimum, or neither by using the first derivative test. This involves analyzing the sign of f'(x) around each critical point.
Choose test points in the intervals determined by the critical points: for example, pick a point less than 0, between 0 and 1, and greater than 1. Evaluate the sign of f'(x) at these test points to determine the behavior of f(x).
If f'(x) changes from positive to negative at a critical point, f has a local maximum there. If f'(x) changes from negative to positive, f has a local minimum. If there is no sign change, the critical point is neither a maximum nor a minimum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local maxima or minima. To find them, set the derivative equal to zero and solve for x. In this case, solve f′(x) = x(x − 1) = 0 to find the critical points.
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Critical Points

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, one can infer the behavior of the function. If f′ changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.
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The First Derivative Test: Finding Local Extrema

Local Maximum and Minimum

A function has a local maximum at a point if the function value at that point is greater than at nearby points. Conversely, a local minimum occurs if the function value is less than at nearby points. Identifying these points involves analyzing the critical points and using tests like the First Derivative Test to confirm the nature of each point.
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The Second Derivative Test: Finding Local Extrema
Related Practice
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.

x⁻⁴ + 2x + 3

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Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫tanθ sec²θ dθ = (1/2) sec²θ + C

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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?

Textbook Question

105. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (c) Acceleration equal to zero?

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.

2 - 5 / x²