Week 7: Extreme values and Saddle points, Lagrange multipliers,

Thomas Calculus, 11 th edition, pages: 1027 - 1048

Sec 14.7 Extreme values and Saddle Point

Continuous functions of two variables assume extreme values on closed, bounded domains. We see in this section that we can narrow the search for these extreme values by examining the functions’ first partial derivatives. A function of two variables can assume extreme values only at domain boundary points or at interior domain
points where both first partial derivatives are zero or where one or both of the first partial derivatives fails to exist. However, the vanishing of derivatives at an interior point (a, b) does not always signal the presence of an extreme value. The surface that is the graph of the function might be shaped like a saddle right above (a, b) and cross its tangent plane there.

Sec 14.8 Lagrange Multipliers

Sometimes we need to find the extreme values of a function whose domain is constrained to lie within some particular subset of the plane—a disk, for example, a closed triangular region, or along a curve. In this section, we explore a powerful method for finding extreme values of constrained functions: the method of Lagrange multipliers.