MATH 16B Lecture Notes - Lecture 8: Minimax, Multiple Integral

• School: Berkeley
• Department: Mathematics
• Course: MATH 16B
• Professor: Alexander Paulin

Document Summary

Max/min of f(x, y, z) such that g(x, y, z) = 0. F(x, y, z, lambda) = f(x, y, z) + (lambda)g(x, y, z) Solve all equations when equal to 0 (a, b, c, d) solution --> (a, b, c) potential max/min. Example: determine the maximum possible volume of a cuboid such that surface area is 6. Volume = xyz; surface area = 2xy + 2xz + 2yz = 6 (assume max volume exists and x, y, z > 0) Xyz = f(x, y, z); 2xy + 2xz + 2yz = 6 --> xy + xz + yz -3 = 0 = g(x, y, z) F(x, y, z, lambda) = xyz + lambda(xy) + lambda(xz) + lambda (yz) - lambda(3) Pd(x) = yz + lambda(y) + lambda(z), pd(y) = xz + lambda(x) + lambda(z), pd(z) = xy + lambda(x) + lambda(y), pd(lambda) = xy + xz + yz - 3.