Optimization in Several Variables

14.7 Optimization in several variables Recall that optimization is the process of finding the extreme values of a function. This amounts to finding the highest and lowest points on the graph over a given domain A local extreme value is a value f(a,b) that is a maximum or minimum in some small open disk around (a,b) Fermat'S theorem for functions of one variable States that if F(a) is a local Definition: extreme value, then a is a critical point and thus the tangent line (if it exists) is horizontal at x = a A function f(x,y) has a local Thus, the tangent plane is horizontal if fx(a,b) = fy(a,b) = 0 - that is, if the extremum At p = (a,b) if there equation reduces to z = f(a,b) exists an open disk D(P,r) such that Example 1: • Local maximum: Show that has one critical point. f (x; y) = 11x ¡ 2xy + 2y + 3y 2 f (x; y) · f (a; b) 2 (x; y) 2 D (P ; r) • Local minimum: f (x; y) ¸ f (a; b) (x; y) 2 D (P ; r) Definition: 22x ¡ 2y = 0 • Fx(a,b) = 0 or fx(a,b) does not exist, and • Fy(a,b) = 0 or fy(a,b) does not exist Theorem 1: ¡2x + 4 (11x) + 3 = 0 22x = 2y ¡2x + 44x + 3 = 0 y = 11x 42x + 3 = 0 y=¡ A point p = (a,b) in the domain of f(xy) is called a critical point if: fy (x; y) = ¡2x + 4y + 3 fx (x; y) = 22x ¡ 2y 11 14 P = µ e ¡ 1 11 ;¡ 14 14 ¶ x=¡ 1 14 We know that in one variable, a function F may have a point of inflection rather than A local extremum at a critical point. A similar phenomenon occurs in several variables However, functions have saddle points, a critical point that is neither a local maximum nor a local minimum As in the one-variable case, there is a second derivative test determining the type of a critical point (a,b) of a function f(x,y) in two variables. This test relies on the sign of the discriminant 2 D = D (a; b) = fxx (a; b) fyy (a; b) ¡ fxy (a; b) If f(x,y) has a local minimum or maximum at p = (a,b), then (a,b) is a critical point of f(x,y) Global extrema Often we are interested in finding the minimum or maximum value of a function f on A given domain D. These are called global or absolute extreme values First, we say that a domain D is bounded if there is a number m > 0 such TheOrem 2: that D is contained in a disk of radius m centered at the origin A point P is called: Let P = (a,b) be a critical point of f(x,y). An interior point of D if D contains some open disk D(p,r) centered at P Assume that f(xx), f(yy), f(xy) are a boundary point of D if every disk centered at P contains points in D continuous near P. then and points not in D The boundary is the curve surrounding the domain. The interior consists of i. Local minimum all points in the domain not lying on the boundary curve fxx (a; b) > 0 A domain D is called closed if D contains all its boundary points (like a D>0 closed interval in r). A domain D is called open it every point of D is an ii. Local maximum interior point (like an open interval in r) f (a; b) < 0 First, a continuous function f on a closed, bound interval [a,b] takes on D>0 both a minimum and a maximum value on [a,b]. Second, these extreme values iii. Saddle point occur either at critical points in the interior (a,b) or at the endpoints xx D<0 iv. Test is inconclusive D=0 Theorem 3: Let f(x,y) be a continuous function on a closed, bounded domain D in R^2. then i. F(x,y) takes on both a minimum and A maximum value on D ii. The extreme values occur either at critical points in the interior of D or at points on the boundary of D