Interactive Real Analysis
- part of Examples 6.4.9(b):
Show that the equation 
= 0
has at least one solution in R
Back
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).
= 0
has at least one solution in R
BackThe 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.