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842 Chapter 25 Functions of several variables
The sign of the second derivative, d2 y
dx2, is then used to distinguish between a maximum
pointandaminimumpoint.Theanalysisisverysimilarforafunctionoftwovariables.
Whenconsideringafunctionoftwovariables, f (x,y),weoftenseekmaximumpoints,
minimum points and saddle points. Collectively these are known as stationary points.
Figure 25.1 illustrates a maximum point at A, a minimum point at B and a saddle point
at C. When leaving a maximum point the value of the function decreases; when leaving
a minimum point the value of the function increases. When leaving a saddle point,
the function increases in one direction, axis D on the figure, and decreases in the other
direction, axis Eon the figure.
To locate stationary points we equate both first partial derivatives to zero, that is we
solve
∂f
∂x = 0
∂f
∂y = 0
Stationarypoints are locatedby solving
∂f
∂x = 0
∂f
∂y = 0
Example25.14 Locatethe stationarypoints of
(a) f(x,y)=2x 2 −xy−7y+y 2
(b) f(x,y)=x 2 −6x+4xy+y 2
Solution (a) Thefirst partialderivatives arefound:
∂f
∂x =4x−y
∂f
∂y =−x−7+2y
The stationary points are located by solving ∂f
∂x =0and∂f ∂y = 0 simultaneously,
thatis
4x−y=0
−x+2y−7=0
Solving these equations yields x = 1, y = 4. Hence the function f (x,y) has one
stationary pointand itislocated at(1, 4).
(b) The first partialderivatives arefound:
∂f
∂x =2x−6+4y
∂f
∂y =4x+2y
The first partialderivatives areequated tozero:
2x+4y−6=0
4x+2y=0
Solving the equations simultaneously yieldsx = −1,y = 2. Thus the function has
one stationary pointlocated at (−1,2).