MATH 237 ASSIGNMENT 7: Optimization Fall 2015

Submit your work before the usual time on Friday, November 13th in the usual place.

1. Let f(x, y) = x2 2x + y3 xy2. Find and classify the critical points of f .2. Find the points on the surface z = x2 + y2 that is closest to the point (1, 1, 0).

3. Consider the function f(x, y) = x4 x2 + y2.(a) Find the linear approximation of f near the point (1, 2).

(b) Find the 2nd-degree Taylor approximation near the same point.

(c) Find all critical points of f , and classify each of them as a local maximum point,local minimum point, or saddle point.

(d) Find the maximum and minimum values of f on D = {(x, y)| |x| 12, |y| 1}.

4. Find the maximum and minimum of f(x, y) = x2 y2 on the region x2 + y2 1.5. Use the method of Lagrange multipliers to find the maximum and minimum values of

xy + z2 on the surface x2 + y2 + z2 = 1.

6. Three alleles (alternative versions of a gene) A, B, and O determine the four bloodtypes A (AA or AO), B (BB or BO), O (OO), and AB. The Hardy-Weinberg Lawstates that the proportion of individuals in a population who carry two different allelesis

P = 2pq + 2pr + 2rq

where p, q, and r are the proportions of A, B, and O in the population. Use the factthat p + q + r = 1 to show that P is at most 2

3.

7.* Find the closest distance between the curves y = ex and y = lnx. Hint: thankfully,there is a quick way to do this one. (There had better be this assignment was longenough!)