9
Finding Local Extreme Values
Example Find the local extreme values of
f ( x , y ) xy x 2 y 2 2 x 2 y 4.
Solution The function is defined and differentiable for all x and y and
3. f ( P0 ) is not an extreme lue of the function f if AC B 2 0; 4. Can not be determined if AC B 2 0.
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Note
2 The expression AC B 2 or f xx f yy f xy is called the discriminant or
order partial derivatives in a neighbourhood of the point P0 ( x0 , y0 ), and P0 is a stationary point of the function f.
A f xx ( P0 ),
Let
B f xy ( P0 ), C f yy ( P0 ).
its domain has no boundary points. The function therefore has extreme
values only at the points where fx and fy are simultaneously zero. These leads to
Example Find the local extreme values of
f ( x , y ) xy x 2 y 2 2 x 2 y 4.
Solution (continued) To see if it does so, we calculate f xx 2, The discriminant of f at (-2,-2) is
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Examples for Saddle Point
z z
y
x x
y
Function point (0,0).
xy( x 2 y 2 ) z at the 2 2 x y
Function
point (0,0).
z y 2 y 4 x 2 at the
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Finding Local Extreme Values
Theorem Suppose that both partial derivatives of the function f ( x , y )
exist at the point ( x0 , y0 ) , and ( x0 , y0 ) is an extreme point of the function f. Then
Section 10.2
Extreme Value Problems of Multivariable Functions
1
Unrestricted Extreme Values
Definition Suppose the that the function f ( x , y ) is defined in U ( x0 , y0 ).
f xx 0, giving rise to a local maximum,and upwards if f xx 0, giving
rise to a local minimum. On the other hand, if the discriminant is negative at P0 ( x0 , y0 ), then the surface curves up in some directions and down in others, so we have a saddle point.
1. Interior points where f x f y 0.
2. Interior points where one or both of f x and f y do not exist. Definition An interior point of the domain of a function f ( x , y ) where both fx and fy are zero or where one or both fx and fy do not
exist everywhere. where Therefore, local extreme values can occur only
fx 2x 0
and
f y 2 y 0.
z
The only possibility is the origin, where the value of f is zero. Since f is never
f x y 2 x 2 0, f y x 2 y 2 0,
or
x y 2.
Therefore, the point (-2,-2) is the only point f may take on an extreme value.
10
Finding Local Extreme Values
For example, the function
z x 2 y 2 attains a minimum at the point
(0,0) and z 1 x 2 y 2 attains a maximum at the point (0,0).
The Necessary Condition for An Extreme Value
Example Find the local extreme values of f ( x , y ) x 2 y 2 .
Solution The domain of f is the entire plane ( so there are no
boundary points ) and the partial derivatives f x 2 x and f y 2 y
Hessian of f. It is sometimes easier to remember it in determinant form,
f xx f yy f
2 xy
f xx f xy
f xy f yy
.
The last theorem says that if the discriminant is positive at the point P0 , then the surface curves the same way in all directions: downwards if
exist is a critical point of f.
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Saddle Point
Thus, the only points where a function f ( x , y ) can assume extreme values are critical points and boundary points. As with differentiable functions of a single variable, not every critical point gives rise to a local extremum. A differentiable function of a single variable might have a point of inflection. A differentiable function of two variables might have a saddle point.
Then
1. f ( P0 ) is a minimum value of the function f if A 0 and AC B 2 0; 2. f ( P0 ) is a maximum value of the function f if A 0 and AC B 2 0;
2 f xx f yy f xy ( 2)( 2) (1)2 4 1 3.
f yy 2,
f xy 1.
The combination
f xy 0 and
negative, we see that the origin gives a
local minimum.
x
y
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Sufficient Condition for Extreme Values
Theorem Suppose that function z f ( x , y ) has continuous second
Definition
A differentiable function f ( x , y ) has a saddle point at a
f ( x , y ) f (a , b ) and domain points (x,y)
stationary point (a,b) if in every open disk centered at (a,b) there are domain points (x,y) where where f ( x , y ) f (a , b ). The corresponding point (a,b,f(a,b)) on the surface z f ( x , y ) is called a saddle point of the surface.
3
f ( x , y ) x 2 y 2 , the point (0,0) is a stationary