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\centerline{MATH 135 -- Calculus 1}
\centerline{The Intermediate Value Theorem}
\centerline{October 7, 2019}
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\noindent
{\it Background}
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\noindent
It's our lucky day, because today we get to discuss some \emph{beautiful mathematics}(!)
The Intermediate Value Theorem (IVT) is the following statement:

\begin{Theorem}[IVT]  Let $f(x)$ be a function that is continuous at every $x$ in a closed
interval $[a,b]$.  Then for every $M$ between and $f(a)$ and $f(b)$, there exists at least
one $x = c$ in the interval where $f(c) = M$.  
\end{Theorem}

What this is saying in intuitive terms is that the graph of a function that is continuous
at every point of a closed interval \emph{is one unbroken curve} that can be drawn without
lifting your pen or pencil from the paper.  In the process it goes through every $y$-value
between the $y$-coordinates of the endpoints $(a,f(a))$ and $(b,f(b))$.  This statement
has some important and surprising consequences!   
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\noindent
{\it Questions}
\begin{enumerate}
\item[(1)] An important consequence:  The IVT implies that many equations have solutions(!)
For instance, if $f(x)$ is a continuous function on an interval
$[a,b]$ and $f(a)\cdot f(b) < 0$ (that is, $f(a)$ and $f(b)$ \emph{have opposite signs}),
then $f(x) = 0$ has a root $x = c$ somewhere in $(a,b)$.  (Note: the root cannot be at one of the
endpoints, since then $f(a)\cdot f(b) = 0$. But we are assuming that product is $< 0$.)  
\begin{enumerate}
\item[(a)]  Look at the particular cubic $P(x) = x^3 - 3x^2 + 5x - 7$ first.  Find 
a real number $a$ where $P(a) < 0$ and another $b > a$ where $P(b) > 0$.  Explain
why $P(x)$ is continuous at all $x$, then use the IVT to argue that $P(x) = 0$ has
a root in your interval $[a,b]$.
\item[(b)]  Now consider a general cubic polynomial $P(x) = Ax^3 + Bx^2 + Cx + D$.
Explain why $P(x) = 0$ must have at least one real root.  (Hint:
since $A \ne 0$ we can divide both sides of the equation $Ax^3 + Bx^2 + Cx + D = 0$
by $A$ to get an equation of the form 
$x^3 + \beta x^2 + \gamma x + \delta = 0$, where $\beta = B/A$, $\gamma = C/A$, etc.  
Explain why, no matter what $\beta,\gamma,\delta$ are, you can always find a really big 
$x = b > 0$ where $b^3 + \beta b^2 + \gamma b + \delta > 0$.  Then apply the IVT
and argue that $P(x) = 0$ has a root in $[a,b]$.)   
\item[(c)]  The same reasoning will apply to any polynomial $P(x)$ satisfying
a particular condition on its \emph{degree}, the highest power of $x$ appearing
with a nonzero coefficient.  What is that condition?
\item[(d)]  Show that $s(x) = x^6 - 8x^4 + 10x^2 - 1 = 0$ has {\it six} real roots, using
the IVT, by finding 6 different intervals $[a,b]$ where $s(a)\cdot s(b) < 0$.  You 
can also find approximations to the roots with a graphing calculator, but that 
is not required.       
\end{enumerate}
\item[(2)]  Another important consequence.  Let $f$ be continuous function on 
the interval $[0,1]$ and suppose that $0 \le f(x) \le 1$ for all $x$ in $[0,1]$.
Explain why $f(x) = x$ for some $x$ in $[0,1]$.  (Hint: Draw a picture to see 
why this is expected, then consider the function $f(x) - x$ and apply the IVT.    
\item[(3)]  A related, but surprising, consequence.  Take any map 
and draw a circle on it anywhere.  (See Figure 6 on page 103 of our textbook, 
for example.)  I claim that the there are two distinct points on the earth
corresponding to points on that circle 
where the temperature right now is 
\emph{exactly the same}.  Here's the idea:  If we put in a coordinate
system with the center of the circle at $(0,0)$, then we can measure the 
counterclockwise angle $\theta$
from the positive $x$-axis to any diameter of the circle.  
Let $f(\theta)$ be \emph{the difference between
the temperatures at the two endpoints of the diameter}.  
In other words, if $A$ and $B$ are the two endpoints
as in the figure, then 
$$f(\theta) = (\text{temperature at } A) -  (\text{temperature at } B).$$  
It is reasonable to assume that $f(\theta)$ is a continuous function of 
$\theta$ (Why?).  Then consider what happens as 
$\theta$ increases from $0$ to $\pi$ radians.  
How are the values $f(0)$ and $f(\pi)$ related?  What
does the IVT tell you about $f$ on the interval $[0,\pi]$?
\end{enumerate}

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