- Consider the function f(x) = 2x² - 16x 5 on the interval [2, 6]. Find the value(s) for c that satisfy Rolle's Theorem in the open interval (2, 6).

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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**Problem Statement:**

Consider the function \( f(x) = 2x^2 - 16x - 5 \) on the interval \([2, 6]\). Find the value(s) for \( c \) that satisfy Rolle's Theorem in the open interval \( (2, 6) \).

**Guidance:**

To solve this problem according to Rolle's Theorem, ensure that the function:

1. Is continuous on the closed interval \([2, 6]\).
2. Is differentiable on the open interval \((2, 6)\).
3. Satisfies \( f(2) = f(6) \).

Once these conditions are met, then there exists at least one value \( c \) in the open interval \((2, 6)\) such that \( f'(c) = 0 \).
Transcribed Image Text:**Problem Statement:** Consider the function \( f(x) = 2x^2 - 16x - 5 \) on the interval \([2, 6]\). Find the value(s) for \( c \) that satisfy Rolle's Theorem in the open interval \( (2, 6) \). **Guidance:** To solve this problem according to Rolle's Theorem, ensure that the function: 1. Is continuous on the closed interval \([2, 6]\). 2. Is differentiable on the open interval \((2, 6)\). 3. Satisfies \( f(2) = f(6) \). Once these conditions are met, then there exists at least one value \( c \) in the open interval \((2, 6)\) such that \( f'(c) = 0 \).
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