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\title{Applied Calculus}
\author{Homework 6}
\date{Due in class, November 17, 2015}
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\begin{enumerate}

\item Find the minimum of the function $f(x,y)=x^2-x+y+y^2$.


\item (From MLS, p.~406.) Find the rectangle with the largest area
that can be placed inside the region bounded between the curve
$y=4-x^2$ and the horizontal axis $y=0$.

\begin{enumerate}
\item Solve this problem by formulating it as a maximization problem
for a function of one variable.

\item Solve this problem by using a Lagrange multiplier.

\end{enumerate}

\item Use a Lagrange multiplier to find the point on the unit circle
$x^2+y^2=1$ closest to $(1,2)$.

\item Use Lagrange multipliers to find the maximum and minimum values
of the function
\[
  f(x,y,z) = 3 \, x - y + 2\, z 
\]
subject to the constraints
\[
  x + y - z = 0  \qquad \text{and} \qquad
  x^2 + 3 \, y^2 = 1 \,.
\]


\item Heron's formula states that the area of a triangle with side
lengths $x$, $y$, and $z$ is
\[
  A = \sqrt{s \, (s-x) \, (s-y) \, (s-z)}
\]
where $s$ is the semiperimeter of the triangle, i.e.,
\[
  s = \frac{x+y+z}2 \,.
\]
Use a Lagrange multiplier to show that the triangle with maximum area
for a given perimeter is equilateral.

\end{enumerate}




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