MathDB

<span class='latex-bold'>(a)</span>

Let

a,b \in \mathbb{R}

and

f,g :\mathbb{R}\rightarrow \mathbb{R}

be differentiable functions. Consider the function

h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix}

Prove that

h

is differentiable and find

h'

. \\ \\

<span class='latex-bold'>(b)</span>

Let

n\in \mathbb{N}

n\geq 3

, take

n-1

pairwise distinct real numbers

a_1<a_2<\dots <a_{n-1}

with sum

\sum_{i=1}^{n-1}a_i = 0

, and consider

n-1

functions

f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}

, each of them

n-2

times differentiable over

\mathbb{R}

. Prove that there exists

a\in (a_1,a_{n-1})

and

\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}

, not all zero, such that

\sum_{k=1}^{n-1} \theta_k a_k=\theta a

and, at the same time,

\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)

for all

i\in\{1,2...,n-1\}

. \\ \\ (Sergiu Novac)

Problem Statement