Generalized Rolle and mean-value theorems

Let the function $f(x)$: (1) be defined on the closed interval $[x_{0},x_{n}]$ and have a continuous $(n-1)$-th derivative $f^{(n-1)}(x)$; (2) have an $n$-th derivative $f^{(n)}(x)$ on the interval $(x_{0},x_{n})$; (3) satisfy the following equalities

Prove that there is at least one point $\xi$ in the interval $(x_{0},x_{n})$ such that

Let the function $f(x)$: (1) be defined on the closed interval $[a,b]$ and have a continuous $(p+q)$-th derivative $f^{(p+q)}(x)$; (2) have a $(p+q+1)$-th derivative $f^{(p+q+1)}(x)$ on the interval $(a,b)$; (3) satisfy the following equalities:

and $f(b)=f'(b)=\cdots =f^{(q)}(b)=0$.

Prove that in this case

where $c$ is some point in the interval $(a,b)$.

Prove that if the function $f$ is differentiable $n$ times on the interval $[a ; b]$ and vanishes at $n+1$ points on it, then there exists $\xi \in (a ; b)$ such that $f^{(n)}(\xi ) = 0$.

Prove that if the function $f$ is differentiable $n$ times on the interval $[a ; b]$, $a = x_{1} < x_{2} < \ldots < x_{k} = b$, $x_{i} \in [a ; b]$, $n_{i}$ are natural numbers, $n_{1}+\ldots +n_{k} = n-1$, and $f^{(j_{i})}(x_{i}) = 0, j_{i} = 0,1, \ldots , n_{i}-1$, $i = 1,2, \ldots , k$, then there exists a point $\xi$ on the interval $[a ; b]$ such that $f^{(n)}(\xi ) = 0$.

If a function coincides at $n+1$ points with a polynomial of degree $n-1$, then its $n$th derivative vanishes at an intermediate point.

Let $0<a<b$. If $f(x)$ vanishes at $n+1$ points of the interval $[a,b]$ and if all the zeros of the polynomial with real coefficients $a_0+a_1x+a_2x^2+\cdots +a_nx^n$ are real, then at an interior point $\xi$ of $[a,b]$

Let the differential equation of $n$th order

have $n-1$ solutions $h_1(x), h_2(x), \ldots , h_{n-1}(x)$ such that, for $a<x<b$, we have

If in the interval $a<x<b$ there are $n+1$ zeros of the function $f(x)$, then there is also a point $\xi$ in the interval such that

(This uses the formula giving a Wronskian-type expression built from $h_1, \ldots , h_{n-1}$ and $f$ as an iterated first-order-operator expression analogous to the one used in the solution to 92 above, from the introduction to §5 of Part Seven.)

The conclusion also holds if the assumption is replaced by the weaker assumption that $f(x)$ has the same values at $n+1$ points as a solution of the differential equation (*). (A generalization of 71 where we are dealing with the equation $y^{(n)}=0$.)

Let

The ratio of the two determinants

is equal to the ratio of the two determinants

where $\xi_1, \xi_2, \ldots , \xi_n$ are suitably chosen intermediate points; in fact

Assuming $x_1<x_2<x_3<\cdots <x_n$, we have

where the intermediate points $\xi_1, \xi_2, \ldots , \xi_n$ satisfy the inequalities given in 95.

where $\xi$ is a suitable interior point of the smallest interval that contains the $n$ distinct points $x_1, x_2, \ldots , x_n$.

where either $x<\xi <x+nh$ or $x+nh<\xi <x$. The special case $n=1$ is the ordinary mean value theorem.

(A generalization of the mean-value theorem to systems of functions different from 95.) Let the functions $h_1(x), h_2(x), \ldots , h_{n-1}(x)$ satisfy the inequalities (**) of 93 and let the function $f(x)$ be arbitrary. If $x_1<x_2<x_3<\cdots <x_n$, then there exists a number $\xi$, $x_1<\xi <x_n$, such that

[Mathematical induction; use 89.]

The closed curve $\mathfrak {C}$ without double points is the boundary of the region $\mathfrak {R}$, a part of the $x,y$-plane. The function $f(x,y)$ is continuous in the closure of $\mathfrak {R}$ (the union of $\mathfrak {R}$ and $\mathfrak {C}$) and has continuous partial derivatives of the second order in $\mathfrak {R}$. If $f(x,y)$ vanishes along $\mathfrak {C}$ and also at some interior point of $\mathfrak {R}$, then there exists a point $\xi ,\eta$ in $\mathfrak {R}$ at which