5.8 STATIONARY VALUES OF MANY-VARIABLE FUNCTIONS
P
S
yxB
Figure 5.2 Stationary points of a function of two variables. A minimum
occurs atB, a maximum atPand a saddle point atS.For functions of two variables, such as the one shown, it should be clear that anecessary condition for a stationary point (maximum, minimum or saddle point)
to occur is that
∂f
∂x= 0 and∂f
∂y=0. (5.21)The vanishing of the partial derivatives in directions parallel to the axes is enough
to ensure that the partial derivative in any arbitrary direction is also zero. The
latter can be considered as the superposition of two contributions, one along
each axis; since both contributions are zero, so is the partial derivative in the
arbitrary direction. This may be made more precise by considering the total
differential
df=∂f
∂xdx+∂f
∂ydy.Using (5.21) we see that although the infinitesimal changesdxanddycan be
chosen independently the change in the value of the infinitesimal functiondfis
always zero at a stationary point.
We now turn our attention to determining the nature of a stationary point ofa function of two variables, i.e. whether it is a maximum, a minimum or a saddle
point. By analogy with the one-variable case we see that∂^2 f/∂x^2 and∂^2 f/∂y^2
must both be positive for a minimum and both be negative for a maximum.
However these are not sufficient conditions since they could also be obeyed at
complicated saddle points. What is important for a minimum (or maximum) is
that the second partial derivative must be positive (or negative) inalldirections,
not just in thex-andy- directions.