Skip to main content
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 4.3.83
Chapter 4, Problem 4.3.83

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 2x³ - 3x² + 12

Verified step by step guidance
1
First, find the first derivative of the function f(x) = 2x³ - 3x² + 12. The first derivative, f'(x), is obtained by differentiating each term: f'(x) = 6x² - 6x.
Next, locate the critical points by setting the first derivative equal to zero and solving for x: 6x² - 6x = 0. Factor the equation: 6x(x - 1) = 0, which gives the critical points x = 0 and x = 1.
Now, find the second derivative of the function to apply the Second Derivative Test. Differentiate the first derivative: f''(x) = d/dx(6x² - 6x) = 12x - 6.
Evaluate the second derivative at each critical point. For x = 0, calculate f''(0) = 12(0) - 6 = -6. Since f''(0) < 0, the function has a local maximum at x = 0.
For x = 1, calculate f''(1) = 12(1) - 6 = 6. Since f''(1) > 0, the function has a local minimum at 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:
4m
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 first derivative is zero or undefined. These points are essential for identifying potential local maxima and minima, as they represent locations where the function's slope changes. To find critical points, one must differentiate the function and solve for the values of x that satisfy the condition f'(x) = 0.
Recommended video:
04:50
Critical Points

Second Derivative Test

The Second Derivative Test is a method used to classify critical points as local maxima, local minima, or saddle points. It involves evaluating the second derivative of the function at the critical points. If f''(x) > 0, the point is a local minimum; if f''(x) < 0, it is a local maximum; and if f''(x) = 0, the test is inconclusive.
Recommended video:
06:02
The Second Derivative Test: Finding Local Extrema

Local Maxima and Minima

Local maxima and minima refer to the highest and lowest points in a specific neighborhood of a function's graph. A local maximum is a point where the function value is greater than that of nearby points, while a local minimum is where it is lower. Understanding these concepts is crucial for analyzing the behavior of functions and optimizing values in various applications.
Recommended video:
07:09
The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

49–54. {Use of Tech} Graphing with technology Make a complete graph of the following functions. A graphing utility is useful in locating intercepts, local extreme values, and inflection points.


ƒ(x) = 3x⁴ + 4x³ - 12x²

Textbook Question

Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.


f'(x) > 0, for all x in the domain of f'; f'(-2) and f'(1) do not exist; f"(0) = 0

Textbook Question

{Use of Tech} Newton’s method and curve sketching Use Newton’s method to find approximate answers to the following questions.

Where are the inflection points of f(x) = (9/5)x⁵ - (15/2)x⁴ + (7/3)x³ + 30x² + 1 located?

Textbook Question

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


lim_u→ π/4 (tan u - cot u) / (u - π/4)

Textbook Question

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


lim_x→0⁺ x²ˣ

Textbook Question

Give the antiderivatives of xᵖ . For what values of p does your answer apply?

1
views