1. (5 points) Find all local extrema of the function f(x,y) = 3y2 − 2y3 − 3x2 + 6xy and classify them.
2. (5 points) Using the method of Lagrange multiplier, find the global maximum and minimum of the objective function f(x,y) = x2y subject to the constraint g(x,y) = x2 + y2 − 3 = 0.
3. (5 points) Suppose that, in a population to be sampled used a stratified sampling, all of the H strata have the same variance. Show that the choice of strata sample sizes n1,··· ,nH that minimizes the variance of the stratified sampling estimator for the sample mean is given by proportional allocation. This is a problem in statistics and the problem reduces to:
(1)
subject to ∑nh = n
h=1
In the above expression, H,Nh,N,Sh and n are constant and the expression in (1) is the variance of the sample mean under stratified sampling, The population has size N and it is divided into H strata. Nh is the size of the population in stratum h. Thus, N1+N2+···NH = N. We sample n < N individuals in the population, nh in stratum h, Sh is the variance of the sample mean in stratum h. Solve the minimization problem without assuming Sh are all equal. Show that if Sh = S for all h, then nh = phn, where ph’s sum to one.
