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Omni-MATH / Find all sequences a₀,a₁,…,a_(n) of real numbers such that a_(n)≠0, for which the following statement is true: If f:R→R is an n times differentiable function and x₀<x₁<…<x_(n) are …

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problem

Find all sequences a0,a1,…,ana_{0}, a_{1},\ldots, a_{n} of real numbers such that an≠0a_{n}\neq 0, for which the following statement is true: If f:R→Rf: \mathbb{R}\to\mathbb{R} is an nn times differentiable function and x0<x1<…<xnx_{0}<x_{1}<\ldots <x_{n} are real numbers such that f(x0)=f(x1)=…=f(xn)=0f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0 then there is h∈(x0,xn)h\in (x_{0}, x_{n}) for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]
Plain-text mathematical notation (without MathML)
Find all sequences a₀,a₁,…,a_(n) of real numbers such that a_(n)≠0, for which the following statement is true:

  If f:R→R is an n times differentiable function
  and x₀<x₁<…<x_(n) are real numbers such that
   f(x₀)=f(x₁)=…=f(x_(n))=0 then there is h∈(x₀,x_(n)) for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]
Original LaTeX notation
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true:

  If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function
  and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that
   $f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]

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Source and history

task_id
bf37b2ed-6ad9-5d55-b0bd-45a2539e7a4e
task_key
test--bf37b2ed-6ad9-5d55-b0bd-45a2539e7a4e
task_revision_id
1
config / split
/ test

Official source

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