Intermediate Value Theorem.

Theorem 5.6.1 (first version; Figure VI1 )   Let $ f$ be a function, defined on the interval $ [a,b]$ . If $ f$ is continuous on $ [a,b]$ , then $ f$ achieves any value between $ f(a)$ and $ f(b)$ .

Figure 7: Intermediate Value Theorem.
\begin{figure}\mbox{\subfigure[]{\epsfig{file=IntVal1.eps,height=4cm}}
\qquad
\subfigure[]{\epsfig{file=IntVal2.eps,height=4cm}}
}\end{figure}

Example 5.6.2   Take $ f(x)=x+\sin x -2$ , for $ - \pi \leq x \leq \pi$ (see Figure 8).

Figure: $ f(x)=x+\sin x -2$
\begin{figure}\centering
\mbox{\epsfig{file=xPlusSinxMinus2.eps, height=4.5cm}}\end{figure}

As $ f(-\pi) < 1 < f(\pi)$ , by Theorem 6.1, the equation $ f(x)=1$ has at least one solution in the interval $ [-\pi, \pi ]$ .

Theorem 5.6.3 (second version; Figure VI2 )   Let $ f$ be a function, defined on the interval $ [a,b]$ . If $ f$ is continuous on $ [a,b]$ and if $ f(a)f(b)<0$ , then $ f$ vanishes at least once on $ (a,b)$ (i.e. there exists at least one real number $ c \in (a,b)$ such that $ f(c)=0$ .

Example 5.6.4   Let $ f(x)=x+\sin x +1$ . By Theorem 6.3, there exists at least one number $ c$ between $ \pi$ and $ \pi$ such that

$\displaystyle c+ \sin c +1 =0.$    

Figure 9:
\begin{figure}\centering
\mbox{\epsfig{file=xPlus1PlusSinx.eps,height=5cm}}\end{figure}

Theorem 5.6.5 (generalization; Figure fig VI3)   Let $ f$ be a function, defined on the interval $ (a,b)$ for which the following conditions hold
(i)
the function $ f$ is continuous on $ (a,b)$ ;
(ii)
the function $ f$ has a limit on the right at $ a$ (eventually infinite);
(iii)
the function $ f$ has a limit on the left at $ b$ (eventually infinite);
(iv)
these limts have different signs.
The $ f$ vanishes at least once on $ (a,b)$ (i.e. there exists at least one real number $ c \in (a,b)$ such that $ f(c)=0$ .

Of course this theorem is still valid when the interval of deinition is the whole of $ \mathbb{R}$ .

Corollary 5.6.6   Every polynomial of odd degree has at least one real root.

Proof. Let $ P(x)$ be a polynomial of odd degree. The function $ P$ is continuous on $ \mathbb{R}$ and by Proposition prop poly limit, it has infinite limits at $ -\infty $ and $ +\infty$ , with opposite signs. The corollary is a direct consequence of Theorem 6.5. $ \qedsymbol$

Figure 10:
\begin{figure}\centering
\mbox{\epsfig{file=oddDegPolyn.eps,height=5cm}}\end{figure}

The graph of $ f: x \mapsto x^5-x^4-2*x^2+1$ is diplayed in Figure 10.

Noah Dana-Picard 2007-12-28