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Some Proofs by Continuity

Published online by Cambridge University Press:  01 August 2016

John Costello
John Costello*
Affiliation:
Department of Education, Loughborough University LE11 3TU
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Extract

How do you know that the equation x3 + x = 5 has one real root? Well, the function x3 + x is monotonic increasing: any increase in the value of x produces an increase in x3 + x. In particular, since 13 + 1 = 2 and 23 + 2 = 10 , we know that, for some x between 1 and 2, x3 + x = 5 . The function must “pass through” 5, and, since it is always increasing, it does this exactly once.

Type
Research Article
Information
The Mathematical Gazette , Volume 78 , Issue 482 , July 1994 , pp. 163 - 166
Copyright
Copyright © The Mathematical Association 1994

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References

1. Apostol, T.M., Mathematical Analysis, Addison-Wesley (1963).Google Scholar
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