Unformatted text preview: ; that is, for at least one real number c such that a < c < b, we have f (c) = 0.
The Intermediate Value Theorem is extremely profound; it gets to the heart of what it means to
be a real number, and is one of the most oft used and under appreciated theorems in Mathematics.
With that being said, most students see the result as common sense, since it says, geometrically,
that the graph of a polynomial function cannot be above the x-axis at one point and below the
x-axis at another point without crossing the x-axis somewhere in between. The following example
uses the Intermediate Value Theorem to establish a fact that that most students take for granted.
Many students, and sadly some instructors, will ﬁnd it silly.
Example 3.1.4. Use the Intermediate Value Theorem to establish that 2 is a real number.
Solution. Consider the polynomial function f (x) = x2 − 2. Then f (1) = −1 and f (3) = 7. Since
f (1) and f (3) have diﬀerent signs, the Intermediate Value Theorem guarantees us a real number
c between 1 and 3 with f (...
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