Intermediate Value Theorem

Definition

The Intermediate Value Theorem states that for any continuous function $f$ on the interval $[a, b]$, if $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one value $c$ in the interval $(a, b)$ such that $f(c) = N$. It guarantees the existence of roots within an interval where the function changes sign.

5 Must Know Facts For Your Next Test

1. The theorem applies only to continuous functions over a closed interval.
2. It is often used to prove the existence of roots of polynomial functions within specific intervals.
3. Intermediate Value Theorem does not tell you how to find the exact value of $c$, only that it exists.
4. To apply the theorem, you need to know the values of the function at the endpoints of the interval.
5. A common application involves showing that a polynomial has at least one real root in a given interval.

Review Questions

• What type of functions does the Intermediate Value Theorem apply to?
• How can you use the Intermediate Value Theorem to show that a polynomial has a root in an interval?
• Does the Intermediate Value Theorem provide an exact value for where a root lies?

Related terms

Continuous Function: A function is continuous if there are no breaks, jumps, or holes in its graph over its domain.

Root: A value for which a given function equals zero; also known as a solution or zero of the function.

Polynomial Function: A mathematical expression consisting of variables and coefficients combined using addition, subtraction, multiplication, and non-negative integer exponents.