N-Tuple Root is Root of n-1th Derivatives of Equation but not nth Derivative

Theorem

Let $\EE$ be the equation:

$\map f x = 0$

where $f: \R \to \R$ is a real function.

Let $\xi$ be a multiple root of $\EE$ with a multiplicity of $n$.

Then for all $k \in \N: k < n$:

$\xi$ is a root of the equation $\map {f^{\paren k} } x = 0$

$\xi$ is not a root of the equation $\map {f^{\paren n} } x = 0$

where $f^{\paren k}: \R \to \R$ is the $k$th derivative of $f$ with respect to $x$.

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiple root
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiple root