14.2 Applications

14.6   Example. We know that $\ln$ is continuous on $\mbox{${\mbox{{\bf R}}}^{+}$}$, and that $\ln(2)\leq 1$ $\leq \ln(4)$.(Cf equation (5.78).) It follows that there is a number $e$ in $[2,4]$ such that $\ln(e) = 1.$

14.7   Example. Two points $P,Q$ on a sphere are called antipodal points if $P$ and $Q$ are opposite ends of the same diameter of the sphere. We will consider the surface of the earth to be a sphere of radius $R$. At any fixed time, let $T(p)$ denote the temperature of the earth at the point $p$ on the surface of the earth. (More precisely, let $T(p)$ be the number such that the temperature at $p$ is $T(p)^{\circ}C$). We will show that there are two antipodal points $P,Q$ on the surface of the earth such that $T(P)=T(Q).$ In fact, we will show that there are two antipodal points on the equator with the same temperature. We first introduce a coordinate system so that the center of the earth is at the origin, and the plane of the equator is the $x$-$y$ plane, and the point on the equator passing through the Greenwich meridian is the point $(R,0)$. Then the points on the equator are the points

$$(R\cos (\theta),R \sin (\theta) ) \mbox{ where } \theta \in \mbox{{\bf R}}.$$

Define a function $f: [0,\pi] \to \mbox{{\bf R}}$ by

$$f(\theta) = T(R\cos(\theta),R\sin(\theta))- T(-R\cos(\theta),-R\sin(\theta )).$$

Thus

$$f(0) = T(R,0) - T(-R,0).$$

We suppose that $f$ is a continuous function on $[0,\pi]$. If $f(0) = 0$ then $T(R,0) = T(-R,0)$, so $(R,0)$ and $(-R,0)$ are a pair of antipodal points with the same temperature. Now

$$f(\pi) = T(-R,0) - T(R,0) = -f(0),$$

so if $f(0) \neq 0$ then $f(0)$ and $f(\pi)$ have opposite signs. Hence by the intermediate value property, there is a number $c \in(0,\pi)$ such that $f(c) = 0$, i.e.

$$T(R\cos(c),R\sin(c)) = T(-R\cos(c),-R \sin(c)).$$

Then $(R\cos(c),R(\sin(c))$ and $(-R\cos(c),-R(\sin(c))$ are a pair of antipodal points with the same temperature.   $\diamondsuit$

14.8   Example. Let

$$P = a_0 + a_1X + a_2X^2 + a_3X^3$$

where $a_0,a_1,a_2,$ and $a_3$ are real numbers, and $a_3 \neq 0$. Then there exists some number $r\in\mbox{{\bf R}}$ such that $P(r) = 0$.

Proof:     I will suppose that $P(t) \neq 0$ for all $t\in\mbox{{\bf R}}$ and derive a contradiction. Let

$$Q(x) = \frac{P(x)}{P(-x)} \mbox{ for all }x \in \mbox{{\bf R}}.$$

Since $P(x) \neq 0$ for all $x \in \mbox{{\bf R}}$, $Q$ is continuous on R. We know that

$$\begin{eqnarray*} \lim\{Q(n)\} &=& \lim\{\frac{a_0 + a_1n + a_2n^2 + a_3n^3} {a... ..._0\over n^3} -{a_1\over n^2}+{a_2\over n} - a_3 } \right\} = -1. \end{eqnarray*}$$

Hence $Q(N) < 0$ for some $N\in\mbox{${\mbox{{\bf Z}}}^{+}$}$. Then $P(N)$ and $P(-N)$ have opposite signs, so by the intermediate value property there is a number $r \in [-N,N]$ such that $P(r) = 0$. This contradicts our assumption that $P(t) \neq 0$ for all $t\in\mbox{{\bf R}}$.  $\diamondsuit$

14.9   Exercise. A Let $p(x) = x^3 - 3x + 1.$ Show that there are at least three different numbers $a,b,c$ such that $p(a) = p(b) = p(c) = 0.$

14.10   Exercise. A Three wires $AC,BC,DC$ are joined at a common point $C$.

Let $S$ be the Y-shaped figure formed by the three wires. Prove that at any time there are two points in $S$ with the same temperature.

14.11   Exercise. A Six wires are joined to form the figure $F$ shown in the diagram.

Show that at any time there are three points in $F$ that have the same temperature. To simplify the problem, you may assume that the temperatures at $A$,$B$, $C$, and $D$ are all distinct.