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Is this a possible way to show that f(x) = arctan(2(x-1)) - lnIxI has exactly four real roots?

Is this a possible way to show that f(x) = arctan(2(x-1)) - lnIxI has exactly four real roots?

Advanced Engineering Mathematics
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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Is this a possible way to show that f(x) = arctan(2(x-1)) - lnIxI has exactly four real roots?

Prove that fex)= arctan (2 (x-1)) - In 1x1 has 4 rocts. First we prove that we cannot have more than four roots. By contradiction, let's assume we have five or mavo roots. By Rolle's Thm. the first derivative would have to have four or more roots. The derivative is -4x² +10x-5 and does not have more than two roots, a camad.! 4x³8x² +5x 2 Now, we will prove that we do have four roots. We will use the IVT (Bolzano's Thm). We will start substituting nice numbers in f, waiting for the value to change signs. ·2|-1|-0.25 0.25/ 0.5 | 1.25|2|3|4|5 X-2-1025/0 f(x) Sign Intervals: (-1,-0.25), (0.25, 0.5), (0.5, 4), (4, 5) Hence, fix) has exactly four real roots! -2....-1.32 0.196 0.403 -0.09.. 0.24... 0.4 6.227.0.02 -0... + + + +++ - 1
Transcribed Image Text:Prove that fex)= arctan (2 (x-1)) - In 1x1 has 4 rocts. First we prove that we cannot have more than four roots. By contradiction, let's assume we have five or mavo roots. By Rolle's Thm. the first derivative would have to have four or more roots. The derivative is -4x² +10x-5 and does not have more than two roots, a camad.! 4x³8x² +5x 2 Now, we will prove that we do have four roots. We will use the IVT (Bolzano's Thm). We will start substituting nice numbers in f, waiting for the value to change signs. ·2|-1|-0.25 0.25/ 0.5 | 1.25|2|3|4|5 X-2-1025/0 f(x) Sign Intervals: (-1,-0.25), (0.25, 0.5), (0.5, 4), (4, 5) Hence, fix) has exactly four real roots! -2....-1.32 0.196 0.403 -0.09.. 0.24... 0.4 6.227.0.02 -0... + + + +++ - 1
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