2.6 Using the IVT to Find Roots
Notes
A root of $f$ is a number $c$ with $f(c) = 0$. If $f$ is continuous and $f(a)$ and $f(b)$ have opposite signs, the IVT (with $N = 0$) guarantees a root in $(a, b)$.
- Compute $f$ at the endpoints and check for a sign change.
- Example: $f(x) = x^3 - x - 1$ has $f(1) = -1$ and $f(2) = 5$, so there is a root in $(1, 2)$.
- No sign change does not prove there is no root.
Practice
Which interval must contain a root of $f(x) = x^3 + x - 3$?
$f(1) = -1$ and $f(2) = 7$ have opposite signs. On $[0, 1]$ we get $-3$ and $-1$, no sign change.
Use the IVT on $[0, 1]$ for $f(x) = 2x^3 - 5x + 1$. What do you find?
$f(0) = 1 > 0$ and $f(1) = 2 - 5 + 1 = -2 < 0$. Polynomials are continuous.
A continuous function has $f(2) = -1$ and $f(3) = 4$. What does the IVT guarantee?
The sign change guarantees at least one root, not exactly one and not at a specific place.
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