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(Use Rolle's Theorem) Let $f\in C^n[\alpha, \beta ]$. Suppose that $f$ has a zero of multiplicity $m$ at $\alpha$ and a root of multiplicity $k$ at $\beta$, where $m \geq 1, k \geq 1$ and $m+k-1=n$. Prove that $f^{(n)}$ has at least one zero in $(\alpha, \beta)$.

Not really sure where to start with this one. Thank you in advance for any help/suggestions.

hungryformath
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1 Answers1

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Prove, as usual, that $f^{(j)}$ has $m+k-j$ roots in the interval $[α,β]$, counted with multiplicity.

As example, in the first step you get per Rolle one root of $f'$ inside $(α,β)$ and the multiplicities at $α$ and $β$ reduce to $m-1$ and $k-1$, so that the total count of roots for $f'$ is $$(m-1)+(k-1)+1=m+k-1.$$

Lutz Lehmann
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