Answered step by step
Verified Expert Solution
Link Copied!

Question

1 Approved Answer

Please explain why the first step of the proof is incorrect. Find the first step in the following Proof that is incorrect, and explain in

Please explain why the first step of the proof is incorrect.

image text in transcribed
Find the first step in the following "Proof" that is incorrect, and explain in a short paragraph why the reasoning is incorrect. Statement: The equation 26456+36=64+6236+cos(6) (8) has a positive solution. In other words, there is some positive number 6 satisfying (8}. Proof: By dividing both sides of the Equation (8) by 64 + 62 36 + cos(6) it follows Equation (8) has a solution if 264 562 + 33 64+62 _36+cos(6) 1 (9) does. Let 2 17(9): 26456 +319 (10) 64+6236+cos(6)' From {2.5.4} of the text it follows that cos is a continuous function. By Theorem 2.4.2 (i), (ii) and (iii) it follows that 264 56 + 36 and 64 + 562 + 36 l 005(6) are both continuous functions. By applying Theorem2.4.2 (v) it follows that F is also a continuous function of 6. Then Equation (8} will have a positive solution provided that F(6) : 1 does. This will be established by using Theorem 2.6.1. Begin by noting that if 6 = 1 then 264 562 l 36 = 0 and hence F(1) = 0. While it is a natural idea that that evaluating 264 562 l 36 at 6 = 1 will be useful, finding a value of6 such that 264 562 + 36 > 1 is not that obvious. In such situations it can be useful to evaluate the limit at 6 goes to 00. With this in mind, observe that , 294 592 + 36 _ 6320 64 +62 36+cos(6) 1' 234/64 56/64 + 39/94 1 9330 94/94 + 92/94 _ 39/94 + com/e4 , 2 56-9 + 36/6-3 all{go 1 + 62 363 l (:os(lSl)/64 111119100 2 562 + 36/6-3 I 2. liuzrlg>0 1 l 6'2 36'3 l cos(6)/64 Lete : 1/2. Then there is some K such that |F(6) 2' 3/2 > 1. To summarize, the following facts about F have been shown: . F is continuous at all 6 Z 0 - F(1) : 0 1 such that F((:3) > 1. It then follows from Theorem 2.6.1 that there must be some 6 between 1 and Q such that F(6) = 1

Step by Step Solution

There are 3 Steps involved in it

Step: 1

blur-text-image

Get Instant Access to Expert-Tailored Solutions

See step-by-step solutions with expert insights and AI powered tools for academic success

Step: 2

blur-text-image

Step: 3

blur-text-image

Ace Your Homework with AI

Get the answers you need in no time with our AI-driven, step-by-step assistance

Get Started

Recommended Textbook for

Bicomplex Holomorphic Functions The Algebra, Geometry And Analysis Of Bicomplex Numbers

Authors: M Elena Luna Elizarrarás, Michael Shapiro, Daniele C Struppa, Adrian Vajiac

1st Edition

3319248685, 9783319248684

More Books

Students also viewed these Mathematics questions

Question

What was the positive value of Max Weber's model of "bureaucracy?"

Answered: 1 week ago