Intermediate Value Theorem

Definition

The Intermediate Value Theorem states that for any continuous function $f$ on a closed interval $[a, b]$, if $N$ is any number between $f(a)$ and $f(b)$, then there exists at least one point $c$ in the interval $(a, b)$ such that $f(c) = N$. This theorem guarantees the existence of a solution within the interval under specific conditions.

5 Must Know Facts For Your Next Test

1. The function must be continuous on the closed interval $[a, b]$.
2. The values $f(a)$ and $f(b)$ are endpoints of the function's range over the interval.
3. There must exist some value $N$ between $f(a)$ and $f(b)$.
4. The theorem does not specify how many points or which exact point(s) will satisfy $f(c) = N$, only that at least one such point exists.
5. It is commonly used to show that equations have roots within an interval.

Review Questions

• What condition must be satisfied by the function for the Intermediate Value Theorem to apply?
• How does the Intermediate Value Theorem help in finding roots of functions?
• If \( f(2) = -1 \) and \( f(5) = 3 \), what can we conclude about \( f(x) \) on \( [2,5] \)?

Related terms

Continuity: A property of a function if it is continuous at every point in its domain; intuitively, its graph has no breaks or holes.

Closed Interval: An interval that includes its endpoints, denoted as \([a,b]\).

Root: A solution to the equation \( f(x) = 0 \); where the graph of \( f(x) \) intersects the x-axis.