### 1 The stationary points of a function of two variables

Figure 7 shows a computer generated picture of the surface defined by the function

$z={x}^{3}+{y}^{3}-3x-3y,$ where both $x$ and $y$ take values in the interval $\left[-1.8,1.8\right]$ .

**
Figure 7
**

There are four features of particular interest on the surface. At point
$A$
there is a
local
maximum
, at
$B$
there is a
**
local minimum
**
, and at
$C$
and
$D$
there are what are known as
**
saddle points
**
.

At $A$ the surface is at its greatest height in the immediate neighbourhood. If we move on the surface from $A$ we immediately lose height no matter in which direction we travel. At $B$ the surface is at its least height in the neighbourhood. If we move on the surface from $B$ we immediately gain height, no matter in which direction we travel.

The features at $C$ and $D$ are quite different. In some directions as we move away from these points along the surface we lose height whilst in others we gain height. The similarity in shape to a horse’s saddle is evident.

At each point
$P$
of a
**
smooth
**
surface one can draw a unique plane which touches the surface there. This plane is called the
**
tangent plane
**
at
$P$
. (The tangent plane is a natural generalisation of the tangent line which can be drawn at each point of a smooth curve.) In Figure 7 at each of the points
$A,B,C,D$
the tangent plane to the surface is horizontal at the point of interest. Such points are thus known as
**
stationary points
**
of the function. In the next subsections we show how to locate stationary points and how to determine their nature using partial differentiation of the function
$f\left(x,y\right)$
,

##### Task!

In Figures 8 and 9 what are the features at $A$ and $B$ ?

**
Figure 8
**

**
Figure 9
**

Figure 8 $A$ is a saddle point, $B$ is a local minimum.

Figure 9 $A$ is a local maximum, $B$ is a saddle point.