[SOLVED] MA3475 Homework 1

1. 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,·· ,n_H 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 ∑n_h = n

h=1

In the above expression, H,N_h,N,S_h 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. N_h is the size of the population in stratum h. Thus, N₁+N₂+···N_H = N. We sample n < N individuals in the population, n_h in stratum h, S_h is the variance of the sample mean in stratum h. Solve the minimization problem without assuming S_h are all equal. Show that if S_h = S for all h, then n_h = p_hn, where p_h’s sum to one.