Calculus III Hughes-Hallett Chapter 15 Optimization.

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Presentation transcript:

Calculus III Hughes-Hallett Chapter 15 Optimization

Local Extrema §f has a l local (relative) maximum at the point P 0( (x 0,y 0 )  D f if f(x 0,y 0 )  f(x,y) for all points P(x,y) near P 0 l local (relative) minimum at the point P 0( (x 0,y 0 )  D f 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: l 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).

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: