\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} Suppose you know that $\displaystyle
{f'(x) = {{2 \over {1+x^4}} - {3 \over {4 +x^4}}}}$. Is $f(0) < f(1)$?

\medskip

\noindent {\bf Note} You probably can't write a formula for a function
with this derivative at this time. Here is such a function:
$$f(x)={\sqrt{2}\over 4} \ln\left( {{x^2+\sqrt{2}x+1} \over
{x^2-\sqrt{2}x+1}}\right)+{\sqrt{2}\over 2} \arctan(\sqrt{2} x + 1)
+{\sqrt{2}\over 2} \arctan(\sqrt{2} x -1)\qquad$$
$$\qquad+{3 \over {16}}\ln(x^2-2x+2) -{3 \over 8} \arctan (x-1) -
{3\over{16}} \ln(x^2+2x+2) - {3\over 8} \arctan(x+1)\,.$$
Does knowing this formula help or is studying the derivative easier?
Please make an  {\it indirect} argument, using information about $f'$.









\vfil\eject\end

