Maxima, minima and saddle points for functions of two variables 359
36.3 Procedure to determine
maxima, minima and saddle
points for functions of two
variables
Given z = f (x, y):
(i) determine
∂z
∂x
and
∂z
∂ y
(ii) for stationary points,
∂z
∂x
= 0 and
∂z
∂ y
= 0,
(iii) solve the simultaneous equations
∂z
∂x
= 0 and
∂z
∂ y
= 0 for x and y, which gives the co-ordinates
of the stationary points,
(iv) determine
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 and
∂ 2 z
∂x∂ y
(v) for each of the co-ordinates of the stationary
points, substitute values of x and y into
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2
and
∂
2 z
∂x∂ y
and evaluate each,
(vi) evaluate
∂ 2 z
∂x∂ y
2
for each stationary point,
(vii) substitute the values of
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 and
∂ 2 z
∂x∂ y
into the equation
=
∂ 2 z
∂x∂ y
2
−
∂ 2 z
∂x 2
∂ 2 z
∂ y 2
and evaluate,
(viii) (a) if >0 then the stationary point is a
saddle point.
(b) if <0 and
∂
2 z
∂x 2 < 0, then the stationary
point is a maximum point,
and
(c) if <0 and
∂
2 z
∂x 2 > 0, then the stationary
point is a minimum point.
36.4 Worked problems on maxima,
minima and saddle points for
functions of two variables
Problem 1. Show that the function
z =(x − 1) 2 + (y − 2) 2 has one stationary point only
and determine its nature. Sketch the surface
represented by z and produce a contour map in the
x-y plane.
Following the above procedure:
(i)
∂z
∂x
= 2(x − 1) and
∂z
∂ y
= 2(y − 2)
(ii) 2(x − 1) =0
(1)
2(y − 2) = 0
(2)
(iii) From equations (1) and (2), x = 1 and y = 2, thus
the only stationary point exists at (1, 2).
(iv) Since
∂z
∂x
= 2(x − 1) = 2x − 2,
∂ 2 z
∂x 2 = 2
and since
∂z
∂ y
= 2(y − 2) = 2y − 4,
∂ 2 z
∂ y 2 = 2
and
∂ 2 z
∂x∂ y
=
∂
∂x
∂z
∂ y
=
∂
∂x
(2y − 4) = 0
(v)
∂ 2 z
∂x 2 =
∂ 2 z
∂ y 2 = 2 and
∂ 2 z
∂x∂ y
= 0
(vi)
∂ 2 z
∂x∂ y
2
= 0
(vii) = (0) 2 − (2)(2) = −4
(viii) Since <0 and
∂
2 z
∂x 2 > 0, the stationary point
(1, 2) is a minimum.
The surface z = (x − 1) 2 + (y − 2) 2 is shown in three
dimensions in Fig. 36.7. Looking down towards the
x-y plane from above, it is possible to produce a contour map. A contour is a line on a map which gives
places having the same vertical height above a datum
line (usually the mean sea-level on a geographical map).
36.3 Procedure to determine
maxima, minima and saddle
points for functions of two
variables
Given z = f (x, y):
(i) determine
∂z
∂x
and
∂z
∂ y
(ii) for stationary points,
∂z
∂x
= 0 and
∂z
∂ y
= 0,
(iii) solve the simultaneous equations
∂z
∂x
= 0 and
∂z
∂ y
= 0 for x and y, which gives the co-ordinates
of the stationary points,
(iv) determine
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 and
∂ 2 z
∂x∂ y
(v) for each of the co-ordinates of the stationary
points, substitute values of x and y into
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2
and
∂
2 z
∂x∂ y
and evaluate each,
(vi) evaluate
∂ 2 z
∂x∂ y
2
for each stationary point,
(vii) substitute the values of
∂ 2 z
∂x 2 ,
∂ 2 z
∂ y 2 and
∂ 2 z
∂x∂ y
into the equation
=
∂ 2 z
∂x∂ y
2
−
∂ 2 z
∂x 2
∂ 2 z
∂ y 2
and evaluate,
(viii) (a) if >0 then the stationary point is a
saddle point.
(b) if <0 and
∂
2 z
∂x 2 < 0, then the stationary
point is a maximum point,
and
(c) if <0 and
∂
2 z
∂x 2 > 0, then the stationary
point is a minimum point.
36.4 Worked problems on maxima,
minima and saddle points for
functions of two variables
Problem 1. Show that the function
z =(x − 1) 2 + (y − 2) 2 has one stationary point only
and determine its nature. Sketch the surface
represented by z and produce a contour map in the
x-y plane.
Following the above procedure:
(i)
∂z
∂x
= 2(x − 1) and
∂z
∂ y
= 2(y − 2)
(ii) 2(x − 1) =0
(1)
2(y − 2) = 0
(2)
(iii) From equations (1) and (2), x = 1 and y = 2, thus
the only stationary point exists at (1, 2).
(iv) Since
∂z
∂x
= 2(x − 1) = 2x − 2,
∂ 2 z
∂x 2 = 2
and since
∂z
∂ y
= 2(y − 2) = 2y − 4,
∂ 2 z
∂ y 2 = 2
and
∂ 2 z
∂x∂ y
=
∂
∂x
∂z
∂ y
=
∂
∂x
(2y − 4) = 0
(v)
∂ 2 z
∂x 2 =
∂ 2 z
∂ y 2 = 2 and
∂ 2 z
∂x∂ y
= 0
(vi)
∂ 2 z
∂x∂ y
2
= 0
(vii) = (0) 2 − (2)(2) = −4
(viii) Since <0 and
∂
2 z
∂x 2 > 0, the stationary point
(1, 2) is a minimum.
The surface z = (x − 1) 2 + (y − 2) 2 is shown in three
dimensions in Fig. 36.7. Looking down towards the
x-y plane from above, it is possible to produce a contour map. A contour is a line on a map which gives
places having the same vertical height above a datum
line (usually the mean sea-level on a geographical map).