Math Problems Vault

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.

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Remark.

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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.

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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.

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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 $f'(x)$ .

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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:

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(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.

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