In-class work for 27 June 2012
1.) Use Lagrange multipliers to find the maximum and minimu values of the function subject to the given constraint:
    a.) $f(x,y) = x^2y; x^2+2y^2=6$
    b.) $f(x,y) = e^{xy}; x^3+y^3=16$
    c.) $f(x,y,z) = x^2y^2z^2; x^2+y^2+z^2=1$

2.) Use Lagrange multipliers to prove that the rectangle with maximum area that has a given perimeter $p$ is a square.

3.) The total production $P$ of a product depends on the amount $L$ of labor used and the amount $K$ of capital invested. Consider the Cobb-Douglas model $P(L,K)=bL^{\alpha}K^{1 - \alpha}$ where $b, \alpha > 0$ and $\alpha < 1$. If the cost of a unit of labor is $m$ and the cost of a unit of capital is $n$, and the company can only spend a total of $p$ dollars, then maximizing production $P$ is subject to the constraint $mL+nK=p$. Use Lagrange multipliers to show the maximum production occurs when $L=\dfrac{\alpha p}{m}$ and $K = \frac{(1-\alpha)p}{n}$.