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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 4.3.78
Chapter 4, Problem 4.3.78

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) = 6x² - x³

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First, find the first derivative of the function f(x) = 6x² - x³. The first derivative, f'(x), is obtained by differentiating each term: f'(x) = d/dx(6x²) - d/dx(x³).
Calculate the first derivative: f'(x) = 12x - 3x².
Set the first derivative equal to zero to find the critical points: 12x - 3x² = 0. Factor the equation: 3x(4 - x) = 0, which gives the critical points x = 0 and x = 4.
Next, find the second derivative of the function to apply the Second Derivative Test. Differentiate f'(x) = 12x - 3x² to get f''(x) = 12 - 6x.
Evaluate the second derivative at each critical point: For x = 0, f''(0) = 12 - 6(0) = 12 (positive, indicating a local minimum). For x = 4, f''(4) = 12 - 6(4) = -12 (negative, indicating a local maximum).

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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 f'(x) = 0.
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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 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.
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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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Related Practice
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