MATH 234 Lecture Notes - Lecture 19: Quadratic Function, Taylor Series, Zero Of A Function

• School: UW-Madison
• Department: Mathematics
• Course: MATH 234
• Professor: Betsy Stovall

Document Summary

Example: f(x, y) = x x3 xy fx = 1 3x2 y2 = 0. There are other critical points (points f(x, y) = 0) {(cid:1876) (cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:882) (cid:1877) (cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:882) y = 1 critical points: (0, 1) , (0, 1) Critical points: ((cid:883) (cid:885) , 0) , ( (cid:883) (cid:885) , 0) So, function f(x, y) has 4 critical points. Zero set of f(x,y): f(x, y) = 0 x x3 xy2 = 0 x(1 x2 y2) = 0. Or { x=(cid:882) x(cid:884)+y(cid:884)=(cid:883) f(x, y) = 0 is equivalent x = (cid:2869) (cid:2871) (x, y) > 0 if (x, y) is an inferior point f(x, y) = 0 if (x, y) is a boundary point. So, the critical point ( (cid:883) (cid:885) , 0) is a local max. By the same analysis, the critical point ( (cid:883) (cid:885) , 0) is a local min (x1, y1) (x2, y2) (xn, yn)