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\title{General Mathematics and Computational Science I}
\author{Exercise 16}
\date{November 8, 2005}
\maketitle

\begin{enumerate}

\item Check the conditions for Laplace's method for the function
\[
  f(x) = x-\ln x - 1
\]
which appears in the proof of  Stirling's formula.  I.e., check the
following. 
\begin{enumerate}
\item $f$ is strictly decreasing for $0<x<1$, strictly increasing for
$x>1$, and $f(1)=0$.
\item There are positive constants $b$ and $c$ such that $f(x) \geq
b\,x$ for $x \geq c$.
\item $f(x) = a \, (x-1)^2 + \psi(x) \, (x-1)^3$ where $\psi$ is a
bounded function for $x \in [1-\delta,1+\delta]$ for some $\delta >
0$.  How must $a$ be chosen?
\end{enumerate}

Hint: For (c), use either Taylor's formula with remainder (easy), or
L'h\^ospital's rule for the function
\[
  g(x) = \frac{f(x)}{(x-1)^2} \,.
\]
(more elementary, but longer).

\item (From Lial \emph{et al.}) Solve the following linear programming
problem: Maximize
\[
  z = 4 \, x + 2 \, y
\]
subject to 
\begin{gather*}
  x + y \leq 5 \,, \\
  2 \, x + y \geq 2 \,, \\
  x + 2 \, y \geq 3 \,, \\
  x \geq 0 \,, \\
  y \geq 0 \,.
\end{gather*}

\item (From Lial \emph{et al.}) A farming community owns 2 acres of
land and can afford to devote 40 days a year to harvesting.  Two crops
can be grown in the region, millet and wheat, which produce
$200\,\text{kg}$ and $400\,\text{kg}$ per acre, respectively.  Millet
requires $10$ days per acre to harvest, while wheat requires $30$ days
per acre.  How many acres should be devoted to each crop to maximize
the weight of grain harvested?

\end{enumerate}
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