Calculus III Hughes-Hallett
Chapter 15 Optimization
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 0 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.
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)
.
Optimization in Three Space
(Unconstrained)
Given z = f(x,y) and suppose that at (a,b,c) the
f(a,b) = 0.
Let A  f xx (a, b), B  f xy (a, b), C  f yy (a, b)
2
and D  AC  .B
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).

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.
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):
f x ( x , y)  lg x ( x , y),

f y ( x , y)  lg y ( x , y),

 g ( x , y)  c.
The Lagrange Equation with Two
Constraints.
£(x,y,z,l1,l2) = f(x,y,z) – l1G1(x,y,z) – l2G2(x,y,z)
which implies:
£ x  f x  l1 (G1 ) x  l2 (G2 ) x  0
£
£ y  f y  l1 (G1 ) y  l2 (G2 ) y  0
£l1  f l1  l1 (G1 ) l1  l2 (G2 ) l1  0
£l2  f l2  l1 (G1 ) l2  l2 (G2 ) l2  0
Interpretation of the Lagrange
Multiplier: l
 The value of l 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:
d
 f ( x(c0 ), y0 (c))  l
dc