Examples for Bolzano's Theorem

Example: Show that the equation = 0 has at least one solution in R

Using a computer it is simple enough to draw this function and to see the approximate solution. However, it is even easier to prove that there must be a solution (without specifying where the solution would be).

The function p(x) is an odd-degree polynomial. Therefore:

If c = , then p(x) = , so that there exists A such that p(A) > 0
If c = - , then p(x) = - , so that there exists B such that p(B) < 0

Hence, by Bolzano's theorem there exists a zero of p(x) between the (unknown !) numbers A and B.

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