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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 4.3.87
Chapter 4, Problem 4.3.87

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) = 2x² ln x - 11x²

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First, find the first derivative of the function \( f(x) = 2x^2 \ln x - 11x^2 \). Use the product rule for the term \( 2x^2 \ln x \) and the power rule for \( -11x^2 \).
Set the first derivative equal to zero to find the critical points. Solve the equation for \( x \) to identify potential critical points.
Next, find the second derivative of the function \( f(x) \). Differentiate the first derivative to obtain the second derivative.
Evaluate the second derivative at each critical point found in step 2. This will help determine the concavity at those points.
Apply the Second Derivative Test: If the second derivative at a critical point is positive, the function has a local minimum there. If it is negative, the function has a local maximum. If the second derivative is zero, the test is inconclusive.

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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. 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. If the second derivative at a critical point is positive, the function has a local minimum there; if negative, it has a local maximum. If the second derivative is zero, the test is inconclusive, and further analysis is needed.
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The Second Derivative Test: Finding Local Extrema

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the logarithm to the base e (approximately 2.718). It is defined for positive x values and plays a crucial role in calculus, particularly in functions involving growth and decay. Understanding its properties is vital when differentiating functions that include ln(x), as it affects the behavior of the function and its derivatives.
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Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question

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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Textbook Question

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


lim_x→π (cos x +1 ) / (x - π )²

Textbook Question

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.


ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .

Textbook Question

A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?

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Textbook Question

{Use of Tech} A pursuit curve A man stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr . The path in the xy-plane followed by the man as he pursues the dog is given by the function y = ƒ(x) = s/2 ((x(ˢ⁺¹)/ˢ) /(s+1) - (x(ˢ⁺¹)/ˢ / s-1)) + s/ s² - 1


Select various values of s > 1 and graph this pursuit curve. Comment on the changes in the curve as s increases. <IMAGE>


Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = tan⁻¹ (x/(x²+2))

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