Chapter 5.1

Rolle's Theorem and Descartes' Rule of Signs

[107/100%]

Show

LaTeX

Show LaTeX

Section 1Zeros of Functions, Changes of Sign of Sequences[21/100%]

Problem 5.1

The two sequences

a_0, a_1, a_2, \ldots , a_n \quad \text{and} \quad a_n, a_{n-1}, a_{n-2}, \ldots , a_0

have the same number of indices of change.

Problem 5.2

Omitting terms of a sequence does not increase the number of changes of sign.

Problem 5.3

Inserting zero terms into a sequence does not alter the number of changes of sign. Inserting into a sequence a new term next to an old term with which it agrees in sign again does not alter the number of changes of sign.

Problem 5.4

The sequence

a_0, \quad a_0 + a_1, \quad a_1 + a_2, \quad \ldots , \quad a_{n-1} + a_n, \quad a_n

has not more changes of sign than the sequence $a_0, a_1, a_2, \ldots , a_n$ .

Problem 5.5

Let the infinite sequence $a_0, a_1, a_2, \ldots , a_n, \ldots$ have only a finite number of changes of sign, and denote this number by $C$ . The sequence

a_0, \quad a_0 + a_1, \quad a_0 + 2a_1 + a_2, \quad \ldots , \quad a_0 + \binom {n}{1} a_1 + \binom {n}{2} a_2 + \cdots + a_n, \quad \ldots

formed from it then has also only finitely many and in fact at most $C$ changes of sign.

Remark.

In a rectangular coordinate system mark the points $(0, a_0), (1, a_1), (2, a_2), \ldots , (n, a_n), \ldots$ and join every two successive points by a straight line (whose horizontal projection thus must be $=1$ ). In the resulting figure the indices of change of the sequence $a_0, a_1, a_2, \ldots$ are quite evident. The zeros of a real analytic function $f(x)$ are not quite so evident from the graph $y = f(x)$ . E.g. a double zero cannot be immediately distinguished from a four-fold zero or a three-fold zero from a five-fold zero even in a very accurately drawn graph.

Problem 5.6

In an interval where $\varphi (x) > 0$ the two functions $f(x)$ and $f(x) \varphi (x)$ have the same zeros.

Problem 5.7

If $p_0 > 0, p_1 > 0, p_2 > 0, \ldots$ , then the two sequences

a_0, a_1, a_2, \ldots \quad \text{and} \quad a_0 p_0, a_1 p_1, a_2 p_2, \ldots

have the same indices of change.

Problem 5.8

Let the values $f(a)$ and $f(b)$ be different from $0$ . The interval $a < x < b$ contains an even or an odd number of zeros of $f(x)$ according as $f(a)$ and $f(b)$ have the same or the opposite sign.

Problem 5.9

Let $a_j$ and $a_k$ be different from $0$ . The subsequence $a_j, a_{j+1}, \ldots , a_{k-1}, a_k$ has an even or an odd number of changes of sign according as $a_j$ and $a_k$ have the same or the opposite sign.

Problem 5.10

Let $a$ and $b$ be successive zeros of $f(x)$

[f(a) = f(b) = 0, \quad f(x) \neq 0 \text{ for } a < x < b].

The derivative $f'(x)$ has an odd number of zeros in the interval $a < x < b$ (and thus at least one zero).

Problem 5.11

Let $j+1$ and $k+1$ be successive indices of change of the sequence $a_0, a_1, a_2, \ldots$ . Then the sequence of differences

a_{j+1} - a_j, \quad a_{j+2} - a_{j+1}, \quad \ldots , \quad a_k - a_{k-1}, \quad a_{k+1} - a_k

has an odd number of changes of sign (and thus at least one change of sign).

Problem 5.12

If $f(x)$ has $Z$ zeros in the interval $a, b$ , then $f'(x)$ has not fewer than $Z-1$ zeros there. This is true whether the interval $a, b$ is open, closed or half open; it may even reduce to a single point.

Problem 5.13

If the sequence

a_0, a_1, a_2, \ldots , a_n

has $C$ indices of change, then the sequence

a_1 - a_0, \quad a_2 - a_1, \quad \ldots , \quad a_n - a_{n-1}

has not fewer than $C-1$ indices of change.

Problem 5.14

If $f(x)$ has $Z$ zeros in the finite interval $a < x < b$ , and if one of the two conditions

\operatorname {sign} f(a) = \operatorname {sign} f'(a) \neq 0, \qquad \operatorname {sign} f(b) = -\operatorname {sign} f'(b) \neq 0

is satisfied, then $f'(x)$ has not fewer than $Z$ zeros in $a, b$ . If both conditions are satisfied then $f'(x)$ has not fewer than $Z+1$ zeros.

Problem 5.15

If the finite sequence

a_0, a_1, a_2, \ldots , a_n

has $C$ changes of sign, then the sequence

a_0, \quad a_1 - a_0, \quad a_2 - a_1, \quad \ldots , \quad a_n - a_{n-1}, \quad -a_n

formed from it has not fewer than $C+1$ changes of sign. (With the exception of the obvious trivial case in which all the terms $a_{\nu }$ of the sequence $=0$ .)

Problem 5.16

If $\lim_{x \to +\infty } f(x) = 0$ , then $f'(x)$ has not fewer zeros than $f(x)$ in the interior of the interval $a, +\infty$ . (A similar result to that for $+\infty$ of course also holds for $-\infty$ .)

Problem 5.17

If $\lim_{n \to \infty } a_n = 0$ , then the infinite sequence

a_0, \quad a_1 - a_0, \quad a_2 - a_1, \quad \ldots , \quad a_n - a_{n-1}, \quad \ldots

has more changes of sign than the sequence

a_0, a_1, a_2, \ldots , a_n, \ldots .

Problem 5.18

Let $\alpha$ be real, and let $f(x)$ have $Z$ zeros in the interval $0 < x < \infty$ . Then the function

\alpha f(x) + f'(x)

has at least $Z-1$ zeros there. Moreover it has at least $Z$ zeros if the condition $\lim_{x \to +\infty } e^{\alpha x} f(x) = 0$ is satisfied.

Problem 5.19

Let $\alpha > 0$ , and let the infinite sequence $a_0, a_1, a_2, \ldots , a_n, \ldots$ have $C$ changes of sign. Then the sequence

\alpha a_0, \quad \alpha a_1 - a_0, \quad \alpha a_2 - a_1, \quad \ldots , \quad \alpha a_n - a_{n-1}, \quad \ldots

has at least $C$ changes of sign. Moreover it has at least $C+1$ changes of sign if the condition $\lim_{n \to \infty } a_n \alpha^n = 0$ is satisfied.

Problem 5.20

If the function $f(x)$ has $Z$ zeros in the interval $0 < x < \infty$ , then the function $\int_0^x f(t)\, dt$ has $Z$ or fewer zeros there.

Problem 5.21

If the infinite sequence

a_0, a_1, a_2, \ldots , a_n, \ldots

has $C$ changes of sign, then the sequence

a_0, \quad a_0+a_1, \quad a_0+a_1+a_2, \quad \ldots , \quad a_0+a_1+a_2+\cdots +a_n, \quad \ldots

has $C$ or fewer changes of sign.

Section 2Reversals of Sign of a Function[6/100%]

Problem 5.22

Let the function $f(x)$ be different from $0$ and of constant sign both in a certain neighbourhood of the point $a$ and in a certain neighbourhood of the point $b$ . The interval $a < x < b$ contains an even or an odd number of reversals of sign of the function $f(x)$ , according as $f(a)$ and $f(b)$ have the same sign or opposite signs.

Remark.

We say that the function $f(x)$ is of constant sign in an interval in the following two cases: (1) $f(x) \leq 0$ in the interval, (2) $f(x) \geq 0$ in the interval. Assume that the interval $a < x < b$ can be divided into $R+1$ sub-intervals such that (1) $f(x)$ is not identically zero in any sub-interval, (2) $f(x)$ is of constant sign in each sub-interval, (3) $f(x)$ is of opposite sign in adjacent sub-intervals. Under these three conditions we say that $f(x)$ has $R$ reversals of sign in the interval $a < x < b$ . A zero of an analytic function does or does not induce a reversal of sign (is or is not a crossing point'') according as the multiplicity of the zero is odd or even. However the concept reversal of sign'' can usefully be applied also to certain non-analytic functions.

Problem 5.23

If the function $f(x)$ has $R$ reversals of sign in the interval $0 < x < \infty$ , then the function $\int_0^x f(t)\, dt$ has in this interval $R$ or fewer reversals of sign.

Problem 5.24

If $R$ is the number of reversals of sign and $Z$ the number of zeros of $f(x)$ in the same open interval, then $Z-R$ is a non-negative even number.

Problem 5.25

If $f(a) = f(b) = 0$ and $f(x) \neq 0$ for $a < x < b$ , then in the interval $a < x < b$ there are an odd number of reversals of sign of the derivative $f'(x)$ .

Problem 5.26

(1)(2)

Let $A_1, A_2, \ldots , A_n$ be non-zero real numbers and let $a_1 < a_2 < a_3 < \cdots < a_n$ . One can prove in various cases that the rational function

f(x) = \frac{A_1}{x-a_1} + \frac{A_2}{x-a_2} + \cdots + \frac{A_n}{x-a_n}

has only real zeros, in particular in the following cases:

(1)

$A_1 > 0, \quad A_2 > 0, \quad \ldots , \quad A_{n-1} > 0$ ;

(2)

$A_1 > 0, \quad A_2 > 0, \quad \ldots , \quad A_{k-1} > 0, \quad A_{k+1} > 0, \quad \ldots , \quad A_n > 0,$

A_1 + A_2 + \cdots + A_n < 0, \quad 1 < k < n.

Problem 5.27

The trigonometric polynomial

f(x) = a_0 + a_1 \cos x + a_2 \cos 2x + \cdots + a_n \cos nx

with real coefficients $a_0, a_1, a_2, \ldots , a_n$ has only real zeros if

\left|a_0\right| + \left|a_1\right| + \left|a_2\right| + \cdots + \left|a_{n-1}\right| < a_n.

Section 3First Proof of Descartes' Rule of Signs[14/100%]

Problem 5.28

The polynomials $P(x)$ and $P(\alpha x)$ have the same number of changes of sign if $\alpha$ is positive.

Remark.

By the changes of sign and the indices of change of the polynomial $a_0+a_1x+a_2x^2+\cdots +a_nx^n$ or the power series $a_0+a_1x+a_2x^2+a_3x^3+\cdots$ we mean the changes of sign and indices of change of the finite or the infinite sequence of coefficients $a_0, a_1, a_2, \ldots , a_n$ or $a_0, a_1, a_2, a_3, \ldots$ respectively.

Problem 5.29

Denote the number of changes of sign of the polynomials

\begin{aligned} P(x) & = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n, \\ P(-x) & = a_0 - a_1 x + a_2 x^2 - \cdots + (-1)^n a_n x^n \end{aligned}

by $C^+$ and $C^-$ respectively. We then have

C^+ + C^- \leq n.

Problem 5.30

Let $\alpha > 0$ . Passing from the polynomial

a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n

to the polynomial

\begin{aligned} (\alpha - x)(a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n) & \\ = \alpha a_0 + (\alpha a_1 - a_0) x + (\alpha a_2 - a_1) x^2 + \cdots & - a_n x^{n+1} \end{aligned}

the number of changes of sign increases; in fact it increases by an odd number.

Remark.

For the case $\alpha = 1$ cf. 15.

Problem 5.31

Let $\alpha > 0$ . Passing from the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

to the power series

(\alpha - x)(a_0 + a_1 x + a_2 x^2 + \cdots ) = \alpha a_0 + (\alpha a_1 - a_0) x + (\alpha a_2 - a_1) x^2 + \cdots

the number of changes of sign does not decrease. In fact it increases if the original series $a_0 + a_1 x + a_2 x^2 + \cdots$ converges for $x = \alpha$ .

Problem 5.32

Let $\alpha > 0$ . Passing from the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

to the power series

(\alpha + x)(a_0 + a_1 x + a_2 x^2 + \cdots ) = \alpha a_0 + (\alpha a_1 + a_0) x + (\alpha a_2 + a_1) x^2 + \cdots

the number of changes of sign does not increase.

Problem 5.33

Passing from the power series

a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

to the power series

\frac{a_0 + a_1 x + a_2 x^2 + \cdots }{1-x} = a_0 + (a_0+a_1) x + (a_0+a_1+a_2) x^2 + \cdots

the number of changes of sign does not increase.

Problem 5.34

Let $\alpha > 0$ . Passing from the power series

a_0 + \frac{a_1}{1!} x + \frac{a_2}{2!} x^2 + \frac{a_3}{3!} x^3 + \cdots

to the power series

e^{\alpha x} \left(a_0 + \frac{a_1}{1!} x + \frac{a_2}{2!} x^2 + \cdots \right) = \sum _{n=0}^{\infty } \frac{a_0 \alpha ^n + \binom {n}{1} a_1 \alpha ^{n-1} + \cdots + a_n}{n!} x^n

the number of changes of sign does not increase.

Remark.

Laguerre, op. cit. 1, p. 22. The proof has gaps.

Problem 5.35

Let $p_1, p_2, \ldots , p_n$ be positive numbers. Set

\begin{aligned} a_0 + \frac{a_1 x}{p_1-x} + \frac{a_2 x^2}{(p_1-x)(p_2-x)} + \cdots + \frac{a_n x^n}{(p_1-x)(p_2-x)\cdots (p_n-x)} & \\ = A_0 + A_1 x + A_2 x^2 + \cdots & \end{aligned}

(the power series expansion is convergent for sufficiently small values of $x$ ). The number of changes of sign of the finite sequence $a_0, a_1, a_2, \ldots , a_n$ is not less than the number of changes of sign of the infinite sequence $A_0, A_1, A_2, \ldots$ .

Remark.

Mathematical induction, 31.

Problem 5.36

Let $Z$ be the number of positive zeros of the polynomial $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ and $C$ the number of changes of sign of the sequence of its coefficients. We then have

C - Z \geq 0.

Remark.

Problem 5.37

Continuation of 36. $C-Z$ is an even number.

Problem 5.38

Let the radius of convergence of the power series $a_0 + a_1 x + a_2 x^2 + \cdots$ be $\rho$ , let the number of its zeros in the interval $0 < x < \rho$ be $Z$ and let the number of changes of sign in the sequence of its coefficients be $C$ . Then

Z \leq C.

Hence, in particular: If $C$ is finite then $Z$ is also finite.

Remark.

Besides 31, the theory of functions of a complex variable must be used.

Problem 5.39

The power series

2 - \frac{x}{1\cdot 2} - \frac{x^2}{2\cdot 3} - \frac{x^3}{3\cdot 4} - \cdots

has no zeros in its circle of convergence. (The sequence of its coefficients has one change of sign---37 cannot be extended to power series without some essential additional condition.)

Problem 5.40

If $\rho = \infty$ or if $\rho$ is finite and $a_0+a_1 \rho +a_2 \rho^2+\cdots$ is divergent, then, provided $C$ is finite, $C-Z$ is a non-negative even number.

Problem 5.41

Let the minimum of the real numbers $\xi_1, \xi_2, \ldots , \xi_n$ be denoted by $\xi_{\alpha }$ and their maximum by $\xi_{\omega }$ . The number of zeros of the polynomial

a_0+a_1(x-\xi _1)+a_2(x-\xi _1)(x-\xi _2)+\cdots +a_n(x-\xi _1)(x-\xi _2)\ldots (x-\xi _n)

in the interval $\xi_{\omega } < x < \infty$ is equal to, or less by an even number than, the number of changes of sign of the sequence

a_0, a_1, a_2, \ldots , a_n,

and the number of zeros in the interval $-\infty < x < \xi_{\alpha }$ is equal to, or less by an even number than, the number of changes of sign of the sequence

a_0, -a_1, a_2, -a_3, \ldots , (-1)^n a_n.

(For $\xi_1 = \xi_2 = \cdots = \xi_n$ equivalent to 36, 37.)

Remark.

Section 4Applications of Descartes' Rule of Signs[13/100%]

Problem 5.42

Let $\lambda$ be a positive proper fraction, $n$ a positive integer. The transcendental equation

1 + \frac{x}{1!} + \frac{x^2}{2!} + \cdots + \frac{x^n}{n!} = \lambda e^x

has a single positive root. This increases monotonically to infinity as $n$ tends to infinity.

Problem 5.43

The function $x^{-5} \left(e^x - 1\right)^{-1}$ of the positive variable $x$ tends to $0$ for $x=0$ and for $x=+\infty$ and between these values has a maximum and no minimum.

Problem 5.43.1

Let $R(f)$ denote the number of reversals of sign of the function $f(x)$ in $(0,1)$ and $Z(K_n)$ the number of zeros of the polynomial

K_n(x) = \sum _{v=0}^{n} f\left(\frac{v}{n}\right) \binom {n}{v} x^v (1-x)^{n-v}, \qquad n = 0, 1, 2, \ldots

(see the introduction to problem 144 of Part Two) in $(0,1)$ . Prove that

Z(K_n) \leq R(f).

Problem 5.43.1

Let $R(f)$ denote the number of reversals of sign of the function $f(x)$ in $(0,1)$ and $Z(K_n)$ the number of zeros of the polynomial

K_n(x) = \sum _{v=0}^{n} f\left(\frac{v}{n}\right) \binom {n}{v} x^v (1-x)^{n-v}, \qquad n = 0, 1, 2, \ldots

(see the introduction to problem 144 of Part Two) in $(0,1)$ . Prove that

Z(K_n) \leq R(f).

Problem 5.44

Let the radius of convergence of the power series $a_0+a_1x+a_2x^2+\cdots$ be $\geq 1$ . The number of its zeros in the interval $0 < x < 1$ does not exceed the number of changes of sign of the sequence

a_0, \quad a_0+a_1, \quad a_0+a_1+a_2, \quad \ldots , \quad a_0+a_1+\cdots +a_n, \quad \ldots .

Problem 5.44.1

Let $C$ be the number of changes of sign of the finite sequence

a_0, \quad a_0+a_1, \quad a_0+a_1+a_2, \quad \ldots , \quad a_0+a_1+\cdots +a_n,

set

P(x) = a_0+a_1x+a_2x^2+\cdots +a_nx^n,

and assume that $P(1)=0$ . Then $P(x)$ has at most $C+1$ positive zeros. Show also that the bound $C+1$ can be attained.

Problem 5.44.1

Let $C$ be the number of changes of sign of the finite sequence

a_0, \quad a_0+a_1, \quad a_0+a_1+a_2, \quad \ldots , \quad a_0+a_1+\cdots +a_n,

set

P(x) = a_0+a_1x+a_2x^2+\cdots +a_nx^n,

and assume that $P(1)=0$ . Then $P(x)$ has at most $C+1$ positive zeros. Show also that the bound $C+1$ can be attained.

Problem 5.45

Let $\left(\dfrac {n}{p}\right)$ denote the Legendre (Jacobi) symbol, defined for $p$ an odd prime and $n$ any number not divisible by $p$ as $\left(\dfrac {n}{p}\right) = 1$ if $n$ is a quadratic residue of $p$ and $\left(\dfrac {n}{p}\right) = -1$ if $n$ is a quadratic non-residue of $p$ (Hardy--Wright, p. 68). The equation

\left(\frac{1}{19}\right) x + \left(\frac{2}{19}\right) x^2 + \left(\frac{3}{19}\right) x^3 + \cdots + \left(\frac{18}{19}\right) x^{18} = 0

has only one positive root, namely $x=1$ .

Remark.

The equation is reciprocal so that it is sufficient to investigate the zeros in the interval $0 < x < 1$ with the aid of 44, 33.

Problem 5.46

The equation of degree $162$

\left(\frac{1}{163}\right) x + \left(\frac{2}{163}\right) x^2 + \left(\frac{3}{163}\right) x^3 + \cdots + \left(\frac{162}{163}\right) x^{162} = 0

has exactly $5$ positive roots which are all simple.

Remark.

Investigate the point $x=0.7$ .

Problem 5.47

If the polynomial $a_0+a_1x+\cdots +a_nx^n$ has only real zeros, then $Z=C$ .

Problem 5.48

Let $\nu_1, \nu_2, \ldots , \nu_n$ be integers, $0 \leq \nu_1 < \nu_2 < \cdots < \nu_n$ , and $0 < \alpha_1 < \alpha_2 < \cdots < \alpha_n$ . Then the determinant (a generalization of the Vandermonde determinant, which corresponds to the case $\nu_1=0, \nu_2=1, \ldots , \nu_n=n-1$ )

\begin{vmatrix} \alpha _1^{\nu _1} & \alpha _1^{\nu _2} & \ldots & \alpha _1^{\nu _n} \\ \alpha _2^{\nu _1} & \alpha _2^{\nu _2} & \ldots & \alpha _2^{\nu _n} \\ \multicolumn {4}{c}{\dotfill } \\ \alpha _n^{\nu _1} & \alpha _n^{\nu _2} & \ldots & \alpha _n^{\nu _n} \end{vmatrix} > 0.

Remark.

First prove that it is $\neq 0$ , 36.

Problem 5.49

Let $a_0 \neq 0$ , $a_n \neq 0$ , and assume that $2m$ consecutive coefficients of the polynomial $a_0+a_1x+\cdots +a_nx^n$ vanish, where $m$ is an integer, $m \geq 1$ . Then the polynomial has at least $2m$ non-real zeros.

Problem 5.50

Let the polynomial $P(x)$ have only real zeros, and let $P(0)=1$ and $P(x)$ not a constant. Set

\frac{1}{P(x)} = 1+b_1x+b_2x^2+\cdots +b_nx^n+\cdots .

Then the polynomial $1+b_1x+\cdots +b_{2m}x^{2m}$ has only non-real zeros.

Remark.

Problem 5.51

Let $m$ be an integer, $m \geq 1$ , and

S(x_1, x_2, \ldots , x_n) = \sum x_1^{l_1} x_2^{l_2} \ldots x_n^{l_n},

where the sum $\sum$ is taken over all sets of non-negative integers $l_1, l_2, \ldots , l_n$ for which

l_1+l_2+\cdots +l_n = 2m.

The homogeneous symmetric function $S(x_1, x_2, \ldots , x_n)$ of the $n$ real variables $x_1, x_2, \ldots , x_n$ is positive definite, i.e. $>0$ for all sets of values of $x_1, x_2, \ldots , x_n$ with the exception of the single set $x_1=0, x_2=0, \ldots , x_n=0$ .

Problem 5.52

Let the polynomial $P(x) = x^n + \cdots$ have only positive zeros. In the power series expansion

\frac{1}{P(x)} = \frac{1}{x^n} + \frac{B_n}{x^{n+1}} + \frac{B_{n+1}}{x^{n+2}} + \cdots

all the coefficients $B_n, B_{n+1}, \ldots$ are positive.

Section 5Applications of Rolle's Theorem[26/100%]

Problem 5.53

The derivative of a polynomial has not more non-real zeros than the polynomial itself.

Problem 5.54

Multiple zeros of the derivative of a polynomial with only real zeros are also multiple zeros of the polynomial itself.

Problem 5.55

If a polynomial has only real and simple zeros, then its higher derivatives have the same property, and moreover every zero of the $(\nu +1)$ th derivative lies between two consecutive zeros of the $\nu$ th.

Problem 5.56

The higher derivatives of the function $(1+x^2)^{-1/2}$ have only real and simple zeros; moreover every zero of the $\nu$ th derivative lies between two consecutive zeros of the $(\nu +1)$ th.

Problem 5.57

The higher derivatives of the function $x(1+x^2)^{-1}$ also have the properties described in 56.

Problem 5.58

One may define the Legendre, the Laguerre and the Hermite polynomials by the formulae

P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2-1)^n, \qquad e^{-x} L_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} e^{-x} x^n, \qquad e^{-\frac{x^2}{2}} H_n(x) = \frac{1}{n!} \frac{d^n}{dx^n} e^{-\frac{x^2}{2}}

respectively (problems 84, 99, 100 of Part Six). Deduce from this definition that these polynomials have only real and simple zeros, which lie in the interior of the intervals

(-1,+1), \qquad (0,+\infty ), \qquad (-\infty ,+\infty )

respectively (problems 97, 99, 100 of Part Six).

Problem 5.59

Let $q$ be an integer, $q \geq 2$ . Let

\frac{d^{n-1}}{dx^{n-1}} \left(\frac{x^{q-1}}{1+x^q}\right) = \frac{Q_n(x)}{(1+x^q)^n}, \qquad \frac{d^n}{dx^n} e^{-x^q} = e^{-x^q} R_n(x).

In the complex plane draw $q$ rays from the origin which divide the plane into equal sectors. Let one of these rays be the positive real axis. Show that the zeros of the polynomials $Q_n(x)$ , $R_n(x)$ lie on these $q$ rays and are arranged in the same way on each ray and finally that those which are $\neq 0$ are simple. (The special case $q=2$ is already covered by 57, 58.)

Problem 5.60

Let $\mu , \nu$ be integers satisfying $0 \leq \mu < \mu +\nu \leq n$ . None of the polynomials

a_{\mu } + \binom {\nu }{1} a_{\mu +1} x + \binom {\nu }{2} a_{\mu +2} x^2 + \cdots + \binom {\nu }{\nu -1} a_{\mu +\nu -1} x^{\nu -1} + a_{\mu +\nu } x^{\nu }

has more nonreal zeros than the polynomial

a_0 + \binom {n}{1} a_1 x + \binom {n}{2} a_2 x^2 + \cdots + \binom {n}{n-1} a_{n-1} x^{n-1} + a_n x^n.

(Generalization of 53.)

Problem 5.61

Let $a_1, a_2, \ldots , a_n$ be positive numbers which are not all equal. Let

(x+a_1)(x+a_2)\ldots (x+a_n) = x^n + \binom {n}{1} m_1 x^{n-1} + \binom {n}{2} m_2^2 x^{n-2} + \cdots + m_n^n.

The numbers $m_1, m_2, \ldots , m_n$ are to be determined by taking the appropriate roots and are to be chosen positive. $m_1$ is the arithmetic mean and $m_n$ the geometric mean of $a_1, a_2, \ldots , a_n$ . Show that

m_1 > m_2 > m_3 > \cdots > m_{n-1} > m_n.

Problem 5.62

Let $\alpha$ be real and $P(x)$ a polynomial. The polynomial $\alpha P(x)+P'(x)$ has not more non-real zeros than the polynomial $P(x)$ itself. (Generalization of 53.)

Problem 5.62.1

The polynomials

s_1^n x + s_2^n x^2 + s_3^n x^3 + \cdots + s_n^n x^n, \qquad S_1^n x + S_2^n x^2 + S_3^n x^3 + \cdots + S_n^n x^n

have only real, non-positive, simple zeros. [See definitions in the introduction to Part One, problems 197 and 186, respectively.]

Definitions (Part One, Chapter 4, 3): $s_k^n$ stands for the number of those permutations of $n$ elements that are the products of $k$ disjoint cycles (the Stirling numbers of the first kind); $S_k^n$ stands for the number of different partitions of a set of $n$ elements into $k$ classes, i.e.into $k$ disjoint non-empty subsets whose union is the whole set (the Stirling numbers of the second kind).

Problem 5.62.1

The polynomials

s_1^n x + s_2^n x^2 + s_3^n x^3 + \cdots + s_n^n x^n, \qquad S_1^n x + S_2^n x^2 + S_3^n x^3 + \cdots + S_n^n x^n

have only real, non-positive, simple zeros. [See definitions in the introduction to Part One, problems 197 and 186, respectively.]

Problem 5.63

Assume that the equation

a_0+a_1x+a_2x^2+\cdots +a_nx^n=0

has only real roots. Then the polynomial

a_0 P(x) + a_1 P'(x) + \cdots + a_n P^{(n)}(x)

has not more non-real zeros than the polynomial $P(x)$ itself.

Problem 5.64

P(x) - \frac{P''(x)}{1!} + \frac{P^{(\mathrm{IV})}(x)}{2!} - \frac{P^{(\mathrm{VI})}(x)}{3!} + \cdots

has not more non-real zeros than the polynomial $P(x)$ itself.

Problem 5.65

If the equation

a_0+a_1x+a_2x^2+\cdots +a_nx^n=0

has only real roots, then the following equation

a_0+\frac{a_1}{1!}x+\frac{a_2}{2!}x^2+\cdots +\frac{a_n}{n!}x^n=0

has also only real roots.

Problem 5.65.1

Assume further that $a_0 \neq 0$ and consider the power series

\frac{1}{a_0+a_1x+a_2x^2+\cdots +a_nx^n} = b_0+b_1x+b_2x^2+\cdots .

Then the polynomial

b_0P(x)+b_1P'(x)+b_2P''(x)+\cdots +b_kP^{(k)}(x)+\cdots

has not less non-real zeros than the polynomial $P(x)$ itself.

Problem 5.65.1

Assume further that $a_0 \neq 0$ and consider the power series

\frac{1}{a_0+a_1x+a_2x^2+\cdots +a_nx^n} = b_0+b_1x+b_2x^2+\cdots .

Then the polynomial

b_0P(x)+b_1P'(x)+b_2P''(x)+\cdots +b_kP^{(k)}(x)+\cdots

has not less non-real zeros than the polynomial $P(x)$ itself.

Problem 5.66

Let $P(x)$ be a polynomial of $n$ th degree, and let the real number $\alpha$ lie outside the interval $[-n,0]$ . Then $\alpha P(x)+xP'(x)$ has not more non-real zeros than $P(x)$ . (A generalization of 53 different from 60 and 62.)

Problem 5.67

Let $Q(x)$ be a polynomial whose zeros are all real and lie outside the interval $[0,n]$ . Then the equation

a_0Q(0)+a_1Q(1)x+a_2Q(2)x^2+\cdots +a_nQ(n)x^n=0

has not more non-real roots than the equation

a_0+a_1x+a_2x^2+\cdots +a_nx^n=0.

Problem 5.68

Let $0<q<1$ . The equation

a_0+a_1qx+a_2q^4x^2+\cdots +a_nq^{n^2}x^n=0

has not more non-real roots than the equation

a_0+a_1x+a_2x^2+\cdots +a_nx^n=0.

Problem 5.69

Let $q > 0$ . The equation

2a_0 + (q+q^{-1}) a_1 x + \left(q^{\sqrt{2}}+q^{-\sqrt{2}}\right) a_2 x^2 + \cdots + \left(q^{\sqrt{n}}+q^{-\sqrt{n}}\right) a_n x^n = 0

has not more non-real roots than the equation

a_0+a_1x+a_2x^2+\cdots +a_nx^n=0.

Problem 5.70

If the curve $y=f(x)$ meets a straight line in three distinct points, then between the outermost intersections there lies at least one point of inflection.

Problem 5.71

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.

Problem 5.72

Let $\alpha$ be a real constant, different from $0, 1, 2, \ldots , n-1$ . The difference

(1+x)^{\alpha } - \left(1+\frac{\alpha }{1}x+\frac{\alpha (\alpha -1)}{1.2}x^2+\cdots +\frac{\alpha (\alpha -1)\ldots (\alpha -n+2)}{1.2\ldots (n-1)}x^{n-1}\right)

vanishes for $x=0$ , but at no other point in the interval $(-1,\infty )$ .

Problem 5.73

The remainder of the exponential series

\frac{x^n}{n!}+\frac{x^{n+1}}{(n+1)!}+\frac{x^{n+2}}{(n+2)!}+\cdots

vanishes for $x=0$ , but for no other real value of $x$ .

Problem 5.74

The $n$ th partial sum of the exponential series

1+\frac{x}{1!}+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}

has no real zero or one real zero according as $n$ is even or odd.

Problem 5.75

Let the polynomials $P_1(x), P_2(x), \ldots , P_l(x)$ be $\not\equiv 0$ and of degree $m_1-1, m_2-1, \ldots , m_l-1$ respectively, and let the real constants $a_1, a_2, \ldots , a_l$ be distinct. The function

g(x) = P_1(x)e^{a_1x}+P_2(x)e^{a_2x}+\cdots +P_l(x)e^{a_lx}

has at most $m_1+m_2+\cdots +m_l-1$ real zeros.

Problem 5.76

Let $\alpha_1<\alpha_2<\cdots <\alpha_n$ , $\beta_1<\beta_2<\cdots <\beta_n$ . Then the determinant

\begin{vmatrix} e^{\alpha _1\beta _1} & e^{\alpha _1\beta _2} & \ldots & e^{\alpha _1\beta _n} \\ e^{\alpha _2\beta _1} & e^{\alpha _2\beta _2} & \ldots & e^{\alpha _2\beta _n} \\ \multicolumn {4}{c}{\dotfill } \\ e^{\alpha _n\beta _1} & e^{\alpha _n\beta _2} & \ldots & e^{\alpha _n\beta _n} \end{vmatrix} > 0.

(Generalization of 48.)

Section 6Laguerre's Proof of Descartes' Rule of Signs[12/100%]

Problem 5.77

Let $a_1, a_2, \ldots , a_n, \lambda_1, \lambda_2, \ldots , \lambda_n$ be real constants, $\lambda_1 < \lambda_2 < \lambda_3 < \cdots < \lambda_n$ . Denote by $Z$ the number of real zeros of the entire function

F(x) = a_1 e^{\lambda _1 x} + a_2 e^{\lambda _2 x} + \cdots + a_n e^{\lambda _n x}

and by $C$ the number of changes of sign in the sequence of numbers $a_1, a_2, \ldots , a_n$ . Then $C-Z$ is a non-negative even integer. (A generalization of 36, 37 different from 41: Replace $e^x$ by $x$ .)

Proof. We may assume without loss of generality that $a_1, a_2, \ldots a_n$ are all different from $0$ . That $C-Z$ is even is evident from the fact that as $x \to -\infty$ the term $a_1 e^{\lambda_1 x}$ is the dominant term, and as $x \to +\infty$ the term $a_n e^{\lambda_n x}$ is the dominant term. [8, 9, 37.] That $C-Z \geq 0$ may be proved by mathematical induction; we deduce the relation for $C$ from that for $C-1$ , with the aid of Rolle's theorem [solution 75]. Indeed, if there are no changes of sign, evidently $Z$ is also $=0$ and the statement holds. Assume that it holds in the case that there are $C-1$ changes of sign. $F(x)$ has $C$ changes of sign, $C \geq 1$ ; e.g. let $\alpha +1$ be an index of change, $1 \leq \alpha < n$ , $a_{\alpha }a_{\alpha +1} < 0$ . Choose a number $\lambda$ in the interval $\lambda_{\alpha } < \lambda < \lambda_{\alpha +1}$ and consider the function

\begin{aligned} F^*(x) & = e^{\lambda x} \frac{d[e^{-\lambda x}F(x)]}{dx} \\ & = a_1(\lambda _1-\lambda )e^{\lambda _1 x}+\cdots +a_{\alpha }(\lambda _{\alpha }-\lambda )e^{\lambda _{\alpha }x} \\ & \quad + a_{\alpha +1}(\lambda _{\alpha +1}-\lambda )e^{\lambda _{\alpha +1}x}+\cdots +a_n(\lambda _n-\lambda )e^{\lambda _n x}. \end{aligned}

Let the number of zeros of $F^*(x)$ be $Z^*$ . We have [6, 12]

Z^* \geq Z-1.

Let the number of changes of sign of the sequence of coefficients

\begin{gathered} -a_1(\lambda -\lambda _1), \quad -a_2(\lambda -\lambda _2), \quad \ldots , \quad -a_{\alpha }(\lambda -\lambda _{\alpha }), \\ a_{\alpha +1}(\lambda _{\alpha +1}-\lambda ), \quad \ldots , \quad a_n(\lambda _n-\lambda ) \end{gathered}

be $C^*$ . Clearly

C^* = C-1.

By the assumption of the mathematical induction argument we have

C^* \geq Z^*.

From the three relations above it follows that

C \geq Z,

q.e.d.---Was it necessary to assume from the outset that $a_1, a_2, \ldots , a_n$ are all $\neq 0$ ? Could we not also have chosen $\lambda =\lambda_{\alpha }$ or $\lambda =\lambda_{\alpha +1}$ ?

Problem 5.77.1

The entire function

F(x) = \sum _{k=1}^{n} (-1)^{n-k} \binom {n}{k} k^x

has simple zeros at the points

x = 1, 2, 3, \ldots , n-1

and no other real zeros. (Interpret $k^x$ as $e^{x \log k}$ , with the principal value of $\log k$ .)

Problem 5.77.1

The entire function

F(x) = \sum _{k=1}^{n} (-1)^{n-k} \binom {n}{k} k^x

has simple zeros at the points

x = 1, 2, 3, \ldots , n-1

and no other real zeros. (Interpret $k^x$ as $e^{x \log k}$ , with the principal value of $\log k$ .)

Problem 5.78

Let $\lambda_1 < \lambda_2 < \lambda_3 < \cdots$ , $\lim_{n \to \infty } \lambda_n = \infty$ . In the interior of its region of convergence (a half-plane bounded on the left), the Dirichlet series

a_1 e^{-\lambda _1 x} + a_2 e^{-\lambda _2 x} + \cdots + a_n e^{-\lambda _n x} + \cdots

has not more real zeros than the number of changes of sign of the sequence of coefficients $a_1, a_2, a_3, \ldots , a_n, \ldots$ . (Generalization of 38.)

Problem 5.79

Let $m$ and $n$ be integers and $a_1, a_2, \ldots , a_n, \lambda_1, \lambda_2, \ldots , \lambda_n$ real numbers that satisfy the conditions

m \geq 1; \qquad n \geq 2; \qquad a_{\nu } \neq 0, \quad \nu =1,2,\ldots ,n; \qquad \lambda _1<\lambda _2<\cdots <\lambda _n.

Assume that the polynomial

P(x) = a_1(x-\lambda _1)^m + a_2(x-\lambda _2)^m + \cdots + a_n(x-\lambda _n)^m

does not vanish identically. Show that the number $Z$ of real zeros of $P(x)$ does not exceed the number $C$ of changes of sign of the sequence

a_1, a_2, a_3, \ldots , a_{n-1}, a_n, (-1)^m a_1

[77, 14].

Problem 5.80

Denote the number of reversals of sign of the function $\varphi (\lambda )$ in the interval $0 < \lambda < \infty$ by $R$ , and the number of real zeros of the integral

F(x) = \int _0^{\infty } \varphi (\lambda ) e^{-\lambda x}\, d\lambda

by $Z$ . Then we have $Z \leq R$ .

In the number of zeros $Z$ we naturally include only the zeros lying in the interior of the region of convergence (a half-plane bounded on the left). (Proof not by application of a limiting process to 77 but by appropriate translation of the proof of 77 to the present case!)

Problem 5.81

The integrals

\int _0^{\infty } f(x) x^n\, dx = M_n, \qquad n=0,1,2,3,\ldots ,

assumed convergent, are called the moments of the function $f(x)$ . Consider the case that not all the moments vanish (the definitions and construction of an example with vanishing moments are given in the introductions to problems 139 of Part Two and 153 of Part Three). Assume e.g. $M_{\mu } \neq 0$ and set

$a_n = M_n$ if $M_n \neq 0$ ,
$a_n = -\operatorname {sign} a_{n+1}$ if $M_n=0$ and $n<\mu$ ,
$a_n = -\operatorname {sign} a_{n-1}$ if $M_n=0$ and $n>\mu$ . (The $a_n$ are defined by recursion; first of all the sign of $a_{\mu }$ must be determined.) The number of reversals of sign of $f(x)$ is not less than the number of changes of sign of the sequence $a_0, a_1, a_2, a_3, \ldots$ . (Problem 140 of Part Two is a special case.)

Problem 5.82

The number of positive zeros lying in the interior of the region of convergence does not exceed the number of reversals of sign of the function $\Phi (\lambda ) = \int_0^{\lambda } \varphi (x)\, dx$ . (Analogue of 44.)

Problem 5.83

The number of positive zeros lying in the interior of the region of convergence does not exceed the number of changes of sign of the sequence

a_1, \quad a_1+a_2, \quad a_1+a_2+a_3, \quad \ldots .

(Generalization of 44.) [80.]

Problem 5.83.1

Let $C$ be the number of changes of sign of the finite sequence

a_1, \quad a_1+a_2, \quad a_1+a_2+a_3, \quad \ldots , \quad a_1+a_2+\cdots +a_n,

$\lambda_i$ ( $i=1,2,\ldots ,n$ ) real numbers satisfying

\lambda _1<\lambda _2<\lambda _3<\cdots <\lambda _n,

and

D(x) = a_1 e^{-\lambda _1 x} + a_2 e^{-\lambda _2 x} + \cdots + a_n e^{-\lambda _n x},

where we assume that $D(0)=0$ . Then $D(x)$ has at most $C+1$ real zeros.

(Both the hypothesis and the conclusion are stronger than in 83. Generalization of 44.1.)

Problem 5.83.1

Let $C$ be the number of changes of sign of the finite sequence

a_1, \quad a_1+a_2, \quad a_1+a_2+a_3, \quad \ldots , \quad a_1+a_2+\cdots +a_n,

$\lambda_i$ ( $i=1,2,\ldots ,n$ ) real numbers satisfying

\lambda _1<\lambda _2<\lambda _3<\cdots <\lambda _n,

and

D(x) = a_1 e^{-\lambda _1 x} + a_2 e^{-\lambda _2 x} + \cdots + a_n e^{-\lambda _n x},

where we assume that $D(0)=0$ . Then $D(x)$ has at most $C+1$ real zeros.

(Both the hypothesis and the conclusion are stronger than in 83. Generalization of 44.1.)

Problem 5.84

The number of positive zeros lying within the region of convergence of the factorial series

a_0 + \frac{1!a_1}{x} + \frac{2!a_2}{x(x+1)} + \frac{3!a_3}{x(x+1)(x+2)} + \cdots

does not exceed the number of changes of sign of the sequence

a_0, \quad a_0+a_1, \quad a_0+a_1+a_2, \quad \ldots .

Remark.

It does not even exceed the number of zeros lying within the interval $0<x<1$ of the power series

f(x) = a_0+a_1x+a_2x^2+\cdots ; \qquad 80, 44.

Problem 5.85

Let the coefficients $p_0, p_1, p_2, \ldots$ of the non-terminating infinite series

F(x) = p_0+p_1x+p_2x^2+\cdots

be non-negative, and let its radius of convergence be $\rho$ . Let $a_1, a_2, \ldots , a_n, \alpha_1, \alpha_2, \ldots , \alpha_n$ be real and

0<\alpha _1<\alpha _2<\alpha _3<\cdots <\alpha _n \leq 1.

The number of zeros of the function

a_1F(\alpha _1x)+a_2F(\alpha _2x)+\cdots +a_nF(\alpha _nx)

in the interval $0<x<\rho$ is not larger than the number of changes of sign of the sequence

a_n, \quad a_n+a_{n-1}, \quad a_n+a_{n-1}+a_{n-2}, \quad \ldots , \quad a_n+a_{n-1}+\cdots +a_2+a_1 \qquad [38, 83].

Problem 5.86

If $0 < \beta_1 < \beta_2 < \cdots < \beta_n$ and $\alpha_n\beta_n$ lies in the interior of the circle of convergence of $F(x)$ , then the determinant

\begin{vmatrix} F(\alpha _1\beta _1) & F(\alpha _1\beta _2) & \ldots & F(\alpha _1\beta _n) \\ F(\alpha _2\beta _1) & F(\alpha _2\beta _2) & \ldots & F(\alpha _2\beta _n) \\ \multicolumn {4}{c}{\dotfill } \\ F(\alpha _n\beta _1) & F(\alpha _n\beta _2) & \ldots & F(\alpha _n\beta _n) \end{vmatrix} \neq 0.

(76 readily follows from this.)

Section 7What is the Basis of Descartes' Rule of Signs?[5/100%]

Problem 5.87

Let the sequence of functions

h_1(x), h_2(x), h_3(x), \ldots , h_n(x)

obey Descartes' rule of signs in the open interval $a<x<b$ . More precisely: If $a_1, a_2, \ldots , a_n$ denote any real numbers which are not all zero, then the number of zeros lying in $a<x<b$ of the linear combination

a_1h_1(x)+a_2h_2(x)+\cdots +a_nh_n(x)

never exceeds the number of changes of sign of the sequence

a_1, a_2, \ldots , a_n.

For this to hold, the following property of the sequence $h_1(x), h_2(x), \ldots , h_n(x)$ is a necessary condition: If $\nu_1, \nu_2, \ldots , \nu_l$ denote integers with $1 \leq \nu_1 < \nu_2 < \nu_3 < \cdots < \nu_l \leq n$ , then the Wronskian determinants (the determinant $W[g_1, \ldots , g_l](x)$ formed from $l$ functions $g_1, \ldots , g_l$ and their derivatives up to order $l-1$ , as defined in the introduction to §5 of Part Seven)

W[h_{\nu _1}(x), h_{\nu _2}(x), h_{\nu _3}(x), \ldots , h_{\nu _l}(x)]

do not vanish in the interval $(a,b)$ and further any two Wronskian determinants with the same number $l$ of rows have the same sign, where $l=1,2,3,\ldots ,n-1$ . [Look at multiple zeros!]

Remark.

We see from problems 36, 41, 77, 84, 85 of this Part that the sequences of functions $1, x, x^2, x^3, \ldots$ ; $1, x-\xi_1, (x-\xi_1)(x-\xi_2), \ldots$ ; $e^{\lambda_1x}, e^{\lambda_2x}, e^{\lambda_3x}, \ldots$ ; $1, 1/x, 1/[x(x+1)], 1/[x(x+1)(x+2)], \ldots$ ; and $F(\alpha_1x), F(\alpha_2x), F(\alpha_3x), \ldots$ considered there have a common property: the number of zeros lying in a certain interval of their linear combinations with constant coefficients never exceeds the number of changes of sign of these coefficients. What is the basis for this frequent validity of Descartes' rule of signs?

Problem 5.88

In particular for the validity of Descartes' rule of signs it is necessary that in the interval $a<x<b$ the quotients

\frac{h_2(x)}{h_1(x)}, \quad \frac{h_3(x)}{h_2(x)}, \quad \ldots , \quad \frac{h_n(x)}{h_{n-1}(x)}

are all positive and are either all monotonically decreasing or all monotonically increasing.

Problem 5.89

Let $1 \leq \alpha \leq n$ . If $h_1(x), h_2(x), \ldots , h_n(x)$ satisfy the determinantal conditions stated in 87, then so do the $n-1$ functions

H_1 = -\frac{d}{dx}\frac{h_1}{h_{\alpha }}, \quad H_2 = -\frac{d}{dx}\frac{h_2}{h_{\alpha }}, \ldots , \quad H_{\alpha -1} = -\frac{d}{dx}\frac{h_{\alpha -1}}{h_{\alpha }},

H_{\alpha } = \frac{d}{dx}\frac{h_{\alpha +1}}{h_{\alpha }}, \ldots , \quad H_{n-2} = \frac{d}{dx}\frac{h_{n-1}}{h_{\alpha }}, \quad H_{n-1} = \frac{d}{dx}\frac{h_n}{h_{\alpha }}.

(The Wronskian expansion formula used here, $W[g_1,\ldots ,g_l] = g_{\alpha }^{l+1} W\left[-(g_1/g_\alpha )', \ldots , (g_l/g_\alpha )'\right]$ up to sign and omission of the index $\alpha$ , is given in the introduction to §5 of Part Seven.)

Remark.

Problem 58 of Part Seven.

Problem 5.90

Let us assume that the Descartes rule of signs is valid for a system of any $n-1$ functions that satisfy the determinantal conditions. Let the function

F(x) = a_1h_1(x)+a_2h_2(x)+\cdots +a_nh_n(x)

have $Z$ zeros in the interior of $a,b$ and let the sequence of coefficients $a_1,a_2,\ldots ,a_n$ have $C$ changes of sign. The criterion given in 87 as a necessary condition for the validity of Descartes' rule of signs is also a sufficient condition [77].

Problem 5.91

Verify that the criterion 87 is satisfied by the functions $e^{\lambda_1x}, e^{\lambda_2x}, \ldots , e^{\lambda_nx}$ , considered in 77.

Section 8Generalizations of Rolle's Theorem[10/100%]

Problem 5.92

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 $[a,b]$

a_0f(\xi )+a_1f'(\xi )+a_2f''(\xi )+\cdots +a_nf^{(n)}(\xi )=0. \qquad [63.]

Problem 5.93

Let the differential equation of $n$ th order

y^{(n)} + \varphi _1(x)y^{(n-1)} + \varphi _2(x)y^{(n-2)} + \cdots + \varphi _n(x)y = 0 \tag {*}

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

h_1(x) > 0, \quad \begin{vmatrix} h_1(x) & h_1'(x) \\ h_2(x) & h_2'(x) \end{vmatrix} > 0, \quad \ldots , \quad \begin{vmatrix} h_1(x) & h_1'(x) & \ldots & h_1^{(n-2)}(x) \\ h_2(x) & h_2'(x) & \ldots & h_2^{(n-2)}(x) \\ \multicolumn {4}{c}{\dotfill } \\ h_{n-1}(x) & h_{n-1}'(x) & \ldots & h_{n-1}^{(n-2)}(x) \end{vmatrix} > 0. \tag {**}

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

f^{(n)}(\xi )+\varphi _1(\xi )f^{(n-1)}(\xi )+\varphi _2(\xi )f^{(n-2)}(\xi )+\cdots +\varphi _n(\xi )f(\xi )=0.

(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.)

Problem 5.94

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 $y^{(n)}=0$ .)

Problem 5.94.1

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

\frac{\partial ^2 f}{\partial x^2} + \frac{\partial ^2 f}{\partial y^2} = 0.

Problem 5.94.1

\frac{\partial ^2 f}{\partial x^2} + \frac{\partial ^2 f}{\partial y^2} = 0.

Problem 5.95

Let

x_1 < x_2 < x_3 < \cdots < x_n.

The ratio of the two determinants

\begin{vmatrix} f_1(x_1) & f_1(x_2) & \ldots & f_1(x_n) \\ f_2(x_1) & f_2(x_2) & \ldots & f_2(x_n) \\ \multicolumn {4}{c}{\dotfill } \\ f_n(x_1) & f_n(x_2) & \ldots & f_n(x_n) \end{vmatrix} : \begin{vmatrix} \varphi _1(x_1) & \varphi _1(x_2) & \ldots & \varphi _1(x_n) \\ \varphi _2(x_1) & \varphi _2(x_2) & \ldots & \varphi _2(x_n) \\ \multicolumn {4}{c}{\dotfill } \\ \varphi _n(x_1) & \varphi _n(x_2) & \ldots & \varphi _n(x_n) \end{vmatrix}

is equal to the ratio of the two determinants

\begin{vmatrix} f_1(\xi _1) & f_1'(\xi _2) & \ldots & f_1^{(n-1)}(\xi _n) \\ f_2(\xi _1) & f_2'(\xi _2) & \ldots & f_2^{(n-1)}(\xi _n) \\ \multicolumn {4}{c}{\dotfill } \\ f_n(\xi _1) & f_n'(\xi _2) & \ldots & f_n^{(n-1)}(\xi _n) \end{vmatrix} : \begin{vmatrix} \varphi _1(\xi _1) & \varphi _1'(\xi _2) & \ldots & \varphi _1^{(n-1)}(\xi _n) \\ \varphi _2(\xi _1) & \varphi _2'(\xi _2) & \ldots & \varphi _2^{(n-1)}(\xi _n) \\ \multicolumn {4}{c}{\dotfill } \\ \varphi _n(\xi _1) & \varphi _n'(\xi _2) & \ldots & \varphi _n^{(n-1)}(\xi _n) \end{vmatrix},

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

\xi _1=x_1, \quad \xi _1<\xi _2<x_2, \quad \xi _2<\xi _3<x_3, \quad \ldots , \quad \xi _{n-1}<\xi _n<x_n.

Problem 5.96

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

\begin{vmatrix} f_1(x_1) & f_1(x_2) & \ldots & f_1(x_n) \\ f_2(x_1) & f_2(x_2) & \ldots & f_2(x_n) \\ \multicolumn {4}{c}{\dotfill } \\ f_n(x_1) & f_n(x_2) & \ldots & f_n(x_n) \end{vmatrix} = \begin{vmatrix} 1 & 1 & \ldots & 1 \\ \binom {x_1}{1} & \binom {x_2}{1} & \ldots & \binom {x_n}{1} \\ \multicolumn {4}{c}{\dotfill } \\ \binom {x_1}{n-1} & \binom {x_2}{n-1} & \ldots & \binom {x_n}{n-1} \end{vmatrix} \cdot \begin{vmatrix} f_1(\xi _1) & f_1'(\xi _2) & \ldots & f_1^{(n-1)}(\xi _n) \\ f_2(\xi _1) & f_2'(\xi _2) & \ldots & f_2^{(n-1)}(\xi _n) \\ \multicolumn {4}{c}{\dotfill } \\ f_n(\xi _1) & f_n'(\xi _2) & \ldots & f_n^{(n-1)}(\xi _n) \end{vmatrix},

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

Problem 5.97

\sum _{\nu =1}^{n} \frac{f(x_{\nu })}{(x_{\nu }-x_1)\ldots (x_{\nu }-x_{\nu -1})(x_{\nu }-x_{\nu +1})\ldots (x_{\nu }-x_n)} = \frac{f^{(n-1)}(\xi )}{(n-1)!},

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

Problem 5.98

f(x+nh) - \binom {n}{1}f(x+(n-1)h) + \binom {n}{2}f(x+(n-2)h) - \cdots + (-1)^nf(x) = h^nf^{(n)}(\xi ),

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

Problem 5.99

(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

\operatorname {sign} \begin{vmatrix} h_1(x_1) & h_1(x_2) & \ldots & h_1(x_n) \\ h_2(x_1) & h_2(x_2) & \ldots & h_2(x_n) \\ \multicolumn {4}{c}{\dotfill } \\ h_{n-1}(x_1) & h_{n-1}(x_2) & \ldots & h_{n-1}(x_n) \\ f(x_1) & f(x_2) & \ldots & f(x_n) \end{vmatrix} = \operatorname {sign} \begin{vmatrix} h_1(\xi ) & h_1'(\xi ) & \ldots & h_1^{(n-1)}(\xi ) \\ h_2(\xi ) & h_2'(\xi ) & \ldots & h_2^{(n-1)}(\xi ) \\ \multicolumn {4}{c}{\dotfill } \\ h_{n-1}(\xi ) & h_{n-1}'(\xi ) & \ldots & h_{n-1}^{(n-1)}(\xi ) \\ f(\xi ) & f'(\xi ) & \ldots & f^{(n-1)}(\xi ) \end{vmatrix}

[Mathematical induction; use 89.]

Problem 5.100

Deduce 76 from 91.