Calculus III HughesHallett Chapter 15 Optimization Local Extrema

Local Extrema § f has a l l local (relative) maximum at the point

Saddle points § A function f, has a saddle point at P 0 if

Optimization in Three Space (Unconstrained) Given z = f(x, y) and suppose that at

Criterion for Global Max/min § Def: A closed region is one which contains its

Constrained Optimization (Lagrange Multipliers) To optimize f(x, y) subject to the constraint g(x, y)

The Lagrange Equation with Two Constraints. £(x, y, z, 1, 2) = f(x, y,

Interpretation of the Lagrange Multiplier: § The value of is the rate of change

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Calculus III Hughes-Hallett Chapter 15 Optimization

Local Extrema § f has a l l local (relative) maximum at the point P 0((x 0, y 0) Df if f(x 0, y 0) f(x, y) for all points P(x, y) near P 0 local (relative) minimum at the point P 0((x 0, y 0) Df if f(x 0, y 0) f(x, y) for all points P(x, y) near P 0 § Points where the gradient is either or undefined are called critical points of the function. If a function as a local max or min at P 0, not on the boundary of its domain, then P 0 is a critical point.

Saddle points § A function f, has a saddle point at P 0 if P 0 is a critical point of f and within any distance of P 0, no matter how small, there are points, P 1 and P 2 with f(P 1) > f(P 0) and f(P 2) < f(P 0).

Optimization in Three Space (Unconstrained) Given z = f(x, y) and suppose that at (a, b, c) the f(a, b) = 0. Let and. Then if: D > 0 and A > 0, then (a, b, c) is a local minimum. l D > 0 and A < 0, then (a, b, c) is a local Maximum. l D < 0 then (a, b, c) is a saddle point. l D = 0, no conclusion can be drawn about (a, b, c). l

Criterion for Global Max/min § Def: A closed region is one which contains its boundary. § Def: A bounded region is one which does not stretch to infinity in any direction. § Criterion: If f is a continuous function on a closed and bounded region R, then f has a global Max at some point (x 0, y 0) in R and a global min at some point (x 1, y 1) in R.

Constrained Optimization (Lagrange Multipliers) To optimize f(x, y) subject to the constraint g(x, y) = c, we can solve the following system of equations for the three unknowns x, y and ( -the Lagrange multiplier):

The Lagrange Equation with Two Constraints. £(x, y, z, 1, 2) = f(x, y, z) – 1 G 1(x, y, z) – 2 G 2(x, y, z) which implies:

Interpretation of the Lagrange Multiplier: § The value of is the rate of change of the optimum value of f as c increases (where g(x, y) = c, or G(x, y) = g(x, y) – c). § If the optimum value of f is written as f(x 0(c), y 0(c)), then we have: