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Some Proofs by Continuity
Published online by Cambridge University Press: 01 August 2016
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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.
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- Copyright © The Mathematical Association 1994
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