For the function given below, which statements concerning Rolle's theorem are true? Select all that apply: f(x) = = x²-x-2 x² - 5x+4 over -6. Rolle's theorem does not apply because f is not continuous on the closed interval [−6, 3]. Rolle's theorem does not apply because of is not differentiable on the open interval (–6, §). Rolle's theorem does not apply because f (-6) ‡ƒ (²}). All three criteria for Rolle's theorem hold, so Rolle's theorem can be used.
For the function given below, which statements concerning Rolle's theorem are true? Select all that apply: f(x) = = x²-x-2 x² - 5x+4 over -6. Rolle's theorem does not apply because f is not continuous on the closed interval [−6, 3]. Rolle's theorem does not apply because of is not differentiable on the open interval (–6, §). Rolle's theorem does not apply because f (-6) ‡ƒ (²}). All three criteria for Rolle's theorem hold, so Rolle's theorem can be used.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Question 7 (Graphical Representation of Concepts)
#### For the function given below, which statements concerning Rolle's theorem are true?
\[ f(x) = \frac{x^2 - x - 2}{x^2 - 5x + 4} \quad \text{over} \quad \left[-6, \frac{5}{3}\right] \]
#### Select all that apply:
1. ☐ Rolle's theorem does not apply because \( f \) is not continuous on the closed interval \(\left[-6, \frac{5}{3}\right]\).
2. ☐ Rolle's theorem does not apply because \( f \) is not differentiable on the open interval \(\left(-6, \frac{5}{3}\right)\).
3. ☐ Rolle's theorem does not apply because \( f(-6) \neq f\left(\frac{5}{3}\right)\).
4. ☐ All three criteria for Rolle's theorem hold, so Rolle's theorem can be used.
#### Explanation:
Rolle’s theorem states that if a function \(f\) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \(f(a) = f(b)\), then there exists at least one \(c\) in the open interval \((a, b)\) such that \(f'(c) = 0\).
In this problem, you are given the rational function \( f(x) = \frac{x^2 - x - 2}{x^2 - 5x + 4} \). You must determine its continuity, differentiability on the specified interval, and check if \( f(-6) = f\left(\frac{5}{3}\right) \).
#### Steps to Solve:
1. **Check Continuity**: Determine if \(f(x)\) is continuous on \(\left[-6, \frac{5}{3}\right]\).
2. **Check Differentiability**: Determine if \(f(x)\) is differentiable on \(\left(-6, \frac{5}{3}\right)\).
3. **Check Function Values**: Calculate \(f(-6)\) and \(f\left(\frac{5}{3}\right)\) to verify if they are equal.
Analyze these conditions to select the correct statements from the provided options.
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Transcribed Image Text:### Question 7 (Graphical Representation of Concepts)
#### For the function given below, which statements concerning Rolle's theorem are true?
\[ f(x) = \frac{x^2 - x - 2}{x^2 - 5x + 4} \quad \text{over} \quad \left[-6, \frac{5}{3}\right] \]
#### Select all that apply:
1. ☐ Rolle's theorem does not apply because \( f \) is not continuous on the closed interval \(\left[-6, \frac{5}{3}\right]\).
2. ☐ Rolle's theorem does not apply because \( f \) is not differentiable on the open interval \(\left(-6, \frac{5}{3}\right)\).
3. ☐ Rolle's theorem does not apply because \( f(-6) \neq f\left(\frac{5}{3}\right)\).
4. ☐ All three criteria for Rolle's theorem hold, so Rolle's theorem can be used.
#### Explanation:
Rolle’s theorem states that if a function \(f\) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \(f(a) = f(b)\), then there exists at least one \(c\) in the open interval \((a, b)\) such that \(f'(c) = 0\).
In this problem, you are given the rational function \( f(x) = \frac{x^2 - x - 2}{x^2 - 5x + 4} \). You must determine its continuity, differentiability on the specified interval, and check if \( f(-6) = f\left(\frac{5}{3}\right) \).
#### Steps to Solve:
1. **Check Continuity**: Determine if \(f(x)\) is continuous on \(\left[-6, \frac{5}{3}\right]\).
2. **Check Differentiability**: Determine if \(f(x)\) is differentiable on \(\left(-6, \frac{5}{3}\right)\).
3. **Check Function Values**: Calculate \(f(-6)\) and \(f\left(\frac{5}{3}\right)\) to verify if they are equal.
Analyze these conditions to select the correct statements from the provided options.
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