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For f(x) = (In x)3 – In x (the same function as in the previous exercise), find the two x-values at which the tangent line to the graph is horizontal. Show how these answers are obtained using techniques of calculus. Approximations to three decimal places are acceptable; however, a bonus is awarded for exact answers.

For f(x) = (In x)3 – In x (the same function as in the previous exercise), find the two x-values at which the tangent line to the graph is horizontal. Show how these answers are obtained using techniques of calculus. Approximations to three decimal places are acceptable; however, a bonus is awarded for exact answers.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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For the function \( f(x) = (\ln x)^3 - \ln x \) (the same function as in the previous exercise), find the two \( x \)-values at which the tangent line to the graph is horizontal. Show how these answers are obtained using techniques of calculus. Approximations to three decimal places are acceptable, however, a bonus is awarded for exact answers.
Transcribed Image Text:For the function \( f(x) = (\ln x)^3 - \ln x \) (the same function as in the previous exercise), find the two \( x \)-values at which the tangent line to the graph is horizontal. Show how these answers are obtained using techniques of calculus. Approximations to three decimal places are acceptable, however, a bonus is awarded for exact answers.
Expert Solution
Step 1

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.

f\left(x\right)=\left(\ln \left(x\right)\right)^3-\ln \left(x^4\right)

take derivative

f'\left(x\right)=\frac{d}{dx}\left(\left(\ln \left(x\right)\right)^3-\ln \left(x^4\right)\right)

f'\left(x\right)=\frac{d}{dx}\left(\left(\ln \left(x\right)\right)^3-\ln \left(x^4\right)\right)

f'(x)= \frac{d}{dx}\left(\left(\ln ^3\left(x\right)\right)\right)-4\frac{d}{dx}\left(\ln \left(x\right)\right)

f'(x)= 3\left(\ln ^2\left(x\right)\right)\cdot \frac{d}{dx}\left(\ln \left(x\right)\right)-4\cdot \frac{1}{x}

f'(x)= 3\left(\ln ^2\left(x\right)\right)\cdot \frac{1}{x}-\frac{4}{x}

f'(x)= \frac{3\ln ^2\left(x\right)-4}{x}

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