Hostname: page-component-5c6d5d7d68-lvtdw Total loading time: 0 Render date: 2024-08-24T00:15:51.607Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Apostol, T.M., Mathematical Analysis, Addison-Wesley (1963).Google Scholar
2. Costello, J., “Quick insights and painted eggs”, Math. Gazette 77, 414–8 (1991).CrossRefGoogle Scholar
3. Poole, B., “Mathematics from soaps”. Mathematics in School, 21, 1 (1992).Google Scholar
4. Selkirk, K., “Mathematics of the kitchen sink – more about minimal points”. Mathematics in School, 21, 5 (1992).Google Scholar
5. See Courant, R. and Robbins, H., What is mathematics?, Oxford (1978).Google Scholar