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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 4.3.94
Chapter 4, Problem 4.3.94

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) = 2x⁻³ - x⁻²

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First, find the first derivative of the function f(x) = 2x⁻³ - x⁻². Use the power rule for derivatives, which states that the derivative of xⁿ is n*xⁿ⁻¹.
The first derivative, f'(x), is calculated as follows: f'(x) = -6x⁻⁴ + 2x⁻³.
Next, find the critical points by setting the first derivative equal to zero and solving for x: -6x⁻⁴ + 2x⁻³ = 0.
Factor the equation to find the values of x that make the derivative zero. This involves factoring out the common term x⁻³: x⁻³(-6x⁻¹ + 2) = 0.
Solve the factored equation for x. The solutions will give the critical points. Then, use the second derivative test to determine the nature of these critical points. Calculate the second derivative, f''(x), and evaluate it at each critical point to determine if they are local maxima or minima.

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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 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 of the first derivative being zero.
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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 the second derivative is positive at a critical point, the function has a local minimum; if negative, it has a local maximum; and if zero, the test is inconclusive.
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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.
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The First Derivative Test: Finding Local Extrema
Related Practice
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