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Q. 57

Found in: Page 249


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Determine whether or not each function f in Exercises 53–60 satisfies the hypotheses of the Mean Value Theorem on the given interval [a, b]. For those that do, use derivatives and algebra to find the exact values of all c ∈ (a, b) that satisfy the conclusion of the Mean Value Theorem.

f(x) = ln(x2+1), [a, b] = [0, 1]

The function satisfies the Mean value theorem and the value of c is 0.4028.

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Step by Step Solution

Step 1. Given information

f(x) = ln(x2+1), [a, b] = [0, 1]

Step 2. Proving Mean Value Theorem. 

The function f(x)=ln (x2+1) is continuous and differentiable on [0,1]. The Mean Value Theorem applies to this function on the interval [0,1].

The slope of the line from (0, f(0)) to (1, f(1)) is:







role="math" localid="1648380951546" c=(2.8855)±(2.8855)24×1×121c=2.8855±4.32612=2.8855±2.082c=2.48275 or 0.40275

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