definition of IVT

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Let $f$ be continuous on the interval $[a, b]$. Let $m = f(a)$ and $M = f(b)$. Let $c$ be a real number between $m$ and $M$ (i.e. either $m \le c \le M$ or $M \le c \le m$). Then there exists a $d$ in $[a, b]$ where $f(d) = c$.

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In layman’s terms, IVT states that if a function $f(x)$ is continuous over an interval $[a,b]$, it will take on every value between $f(a)$ and $f(b)$ somewhere in that interval.

using IVT to find roots of a function

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IVT also means that if $f(a)f(b) < 0$, then $f$ has a root/zero somewhere in the interval $(a,b)$.

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This is because if $f(a)f(b)<0$, then exactly one of $f(a)$ or $f(b)$ is negative and the other is positive, meaning by IVT, $f(x)$ must equal 0 for some $x$ in between $a$ and $b$.

In addition, the root cannot be at $x=a$ or $x=b$, because otherwise $f(a)f(b)$ would equal 0.

• Example: show $f(x) = x^3 - 2$ has a positive root using IVT

IVT can also be used to show that solutions to some equations exist.

• Example: show that there is a real number $x$ such that $\cos(x) = 2x$