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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.1.23
Chapter 4, Problem 4.1.23

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = 3x² - 4x + 2

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To find the critical points of the function \( f(x) = 3x^2 - 4x + 2 \), we first need to find its derivative. The derivative, \( f'(x) \), represents the slope of the tangent line to the curve at any point \( x \).
Differentiate the function \( f(x) = 3x^2 - 4x + 2 \) with respect to \( x \). Using the power rule, the derivative of \( 3x^2 \) is \( 6x \), and the derivative of \( -4x \) is \( -4 \). The derivative of a constant, \( 2 \), is \( 0 \). Therefore, \( f'(x) = 6x - 4 \).
Critical points occur where the derivative is zero or undefined. Since \( f'(x) = 6x - 4 \) is a linear function, it is defined for all \( x \). Set the derivative equal to zero to find the critical points: \( 6x - 4 = 0 \).
Solve the equation \( 6x - 4 = 0 \) for \( x \). Add \( 4 \) to both sides to get \( 6x = 4 \). Then, divide both sides by \( 6 \) to isolate \( x \), resulting in \( x = \frac{4}{6} \). Simplify the fraction to get \( x = \frac{2}{3} \).
The critical point of the function \( f(x) = 3x^2 - 4x + 2 \) is at \( x = \frac{2}{3} \). To confirm whether this point is a maximum, minimum, or neither, you can use the second derivative test or analyze the behavior of \( f'(x) \) around \( x = \frac{2}{3} \).

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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 either zero or undefined. These points are essential for identifying local maxima, minima, and points of inflection. To find critical points, one typically takes the derivative of the function and solves for the values of x that satisfy the condition.
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Critical Points

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental concept in calculus, representing the slope of the tangent line to the curve at any given point. For polynomial functions like ƒ(x) = 3x² - 4x + 2, the derivative can be calculated using power rules.
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Derivatives

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function given, ƒ(x) = 3x² - 4x + 2, is a quadratic polynomial, which has a parabolic graph. Understanding the properties of polynomial functions is crucial for analyzing their behavior, including finding critical points.
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Introduction to Polynomial Functions
Related Practice
Textbook Question

Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.


p(x) = 3 sec² x

Textbook Question

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.


f(x) = 8x³ + sin x; F(0) = 2

Textbook Question

{Use of Tech} Absolute maxima and minima


a. Find the critical points of f on the given interval.

b. Determine the absolute extreme values of f on the given interval.

c. Use a graphing utility to confirm your conclusions.


f(x) = 2ᶻ sin x on [-2,6]

Textbook Question

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.


f(x) = cos⁻¹ x - x; x₀ = 0.75

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Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)

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Textbook Question

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

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