Question 2. Our goal is to prove that f(x)= x + e* has exactly one root (i.e. f(x) = 0 for exactly onevalue of x)."a) First, we really need this function to be both continuous and differentiable. Explain in words howwe know this is true.b) Now, let's prove that f(x) has at least one root. What theorem do we need for that? Hint: Thisis a review problem! We're looking all the way back at section 2.5.c) Use the theorem you stated in part a) to prove that f(x) has at least one root.d) Now that we've proven there is at least one root, we will now prove there is not at least tworoots. We'll do this using contradiction. Let's (incorrectly) assume that there are in fact twodifferent roots, and use that to blow up math. The logic here is that if we assume something, andend up breaking math using that assumption, then our assumption should be wrong and we cancorrectly conclude its opposite.Let's call those two roots a and b. Mean Value Theorem then states that the derivative of f(x)should equal f(b)-f(a) somewhere. What is f(b)-f(a) numerically equal to in this case?e) Calculate f'(x) and find its range. Does your answer from part d) fall in the range? In other words,even though MVT said this must happen, is it actually true that f'(x) can attain the value foundin part d)?

Answered: Image /qna-images/answer/f7cadf47-4bbc-4ba5-94bc-08c923932fbe.jpg

Question 2. Our goal is to prove that f(x) = x + e* has exactly one root (i.e. f(x) = 0 for exactly one value of x)." a) First, we really need this function to be both continuous and differentiable. Explain in words how we know this is true. b) Now, let's prove that f(x) has at least one root. What theorem do we need for that? Hint: This is a review problem! We're looking all the way back at section 2.5. c) Use the theorem you stated in part a) to prove that f(x) has at least one root.

Question 2. Our goal is to prove that f(x) = x + e* has exactly one root (i.e. f(x) = 0 for exactly one value of x)." a) First, we really need this function to be both continuous and differentiable. Explain in words how we know this is true. b) Now, let's prove that f(x) has at least one root. What theorem do we need for that? Hint: This is a review problem! We're looking all the way back at section 2.5. c) Use the theorem you stated in part a) to prove that f(x) has at least one root.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter6: Applications Of The Derivative

Section6.CR: Chapter 6 Review

Problem 1CR

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