Skip to main content
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 4.3.5c
Chapter 4, Problem 4.3.5c

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 − 1)(x + 2)(x − 3)

Verified step by step guidance
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 equal to zero or is undefined. In this case, f'(x) = (x - 1)(x + 2)(x - 3) is a polynomial, so it is defined everywhere. Set f'(x) = 0 to find the critical points.
Solve the equation (x - 1)(x + 2)(x - 3) = 0. This equation is satisfied when any of the factors is zero. Therefore, the critical points are x = 1, x = -2, and x = 3.
Next, determine whether each critical point is a local maximum, local minimum, or neither. This can be done using the First Derivative Test. Evaluate the sign of f'(x) around each critical point.
Choose test points in the intervals determined by the critical points: (-∞, -2), (-2, 1), (1, 3), and (3, ∞). For each interval, pick a test point and substitute it into f'(x) to determine the sign of the derivative in that interval.
Based on the sign changes of f'(x) around each critical point, conclude whether each point is a local maximum, local minimum, or neither. If f'(x) changes from positive to negative at a critical point, it is a local maximum. If f'(x) changes from negative to positive, it is a local minimum. If there is no sign change, the point is neither.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

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, f′(x) = (x − 1)(x + 2)(x − 3) = 0, so the critical points are x = 1, x = -2, and x = 3.
Recommended video:
04:50
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, we can infer the behavior of the function. If f′ changes from positive to negative at a critical point, it's a local maximum; if it changes from negative to positive, it's a local minimum.
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema

Sign Analysis of Derivative

Sign analysis involves examining the intervals between critical points to determine the sign of the derivative. This helps in understanding the function's increasing or decreasing behavior. For f′(x) = (x − 1)(x + 2)(x − 3), evaluate the sign of f′ in intervals like (-∞, -2), (-2, 1), (1, 3), and (3, ∞) to apply the First Derivative Test effectively.
Recommended video:
06:51
Derivatives Applied To Acceleration Example 2
Related Practice
Textbook Question

Checking Antiderivative Formulas


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


∫xsinx dx = -x cos x + C

1
views
Textbook Question

[Technology Exercise] 16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

" style="" width="335">

b. Find the domain of V for the problem situation and graph V over this domain.

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.

x² − 2x + 1

Textbook Question

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[−f(x)] dx

1
views
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.

sin πx − 3sin 3x

1
views
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)²(x + 2)