1. Calculus III Hughes-Hallett Chapter 15 Optimization

2. Local Extrema f has a local (relative) maximum at the point P0((x0,y0) Df if f(x0,y0)  f(x,y) for all points P(x,y) near P0 local (relative) minimum at the point P0((x0,y0) Df if f(x0,y0)  f(x,y) for all points P(x,y) near P0 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 P0, not on the boundary of its domain, then P0 is a critical point.

3. Saddle points A function f, has a saddle point at P0 if P0 is a critical point of f and within any distance of P0, no matter how small, there are points, P1 and P2 with f(P1) > f(P0) and f(P2) < f(P0) .

4. 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. D > 0 and A < 0, then (a,b,c) is a local Maximum. D < 0 then (a,b,c) is a saddle point. D = 0, no conclusion can be drawn about (a,b,c).

5. 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 (x0,y0) in R and a global min at some point (x1,y1) in R.

6. 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 l (l -the Lagrange multiplier):

7. The Lagrange Equation with Two Constraints. £(x,y,z,1,2) = f(x,y,z) – 1G1(x,y,z) – 2G2(x,y,z) which implies:

8. 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(x0(c),y0(c)), then we have: