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Extreme Values of
Multivariate Functions
Professor Peter Cramton
Economics 300
Extreme values of multivariate functions
• In economics many problems reflect a need to
choose among multiple alternatives
– Consumers decide on consumption bundles
– Producers choose a set of inputs
– Policy-makers may choose several instruments to
motivate behavior
• We now generalize the univariate techniques
Stationary points and tangent planes
of bivariate functions
2 2
1 1 2 26 16 4g x x x x 2 2
1 2 1 2 1 24 2 16h x x x x x x
Slices of a bivariate function 2 2
1 1 2 26 16 4g x x x x
1 16 2 0g x 2 216 8 0g x
Multivariate first-order condition
• If is differentiable with respect to each
of its arguments and reaches a maximum or a
minimum at the stationary point, then
each of the partial derivatives evaluated at that
point equals zero, i.e.
1 2( , ,..., )nf x x x
* *
1( ,..., )nx x
* *
1 1
* *
1
( ,..., ) 0
...
...
...
( ,..., ) 0
n
n n
f x x
f x x
Second-order condition in the bivariate
case First total differential
1 2
1 1 2 1 2 1 2 2
1 1 2 2
( , )
( , ) ( , )
i.e.
y f x x
dy f x x dx f x x dx
dy f dx f dx
1 2( , )f x x
Second-order condition in the bivariate
case Second total differential
2
1 2
1 2
1 1 2 2 1 1 2 21 2
1 2
2 2
11 1 12 1 2 22 2
[ ] [ ]
[ ] [ ]
( ) 2 ( ) ( )
dy dyd y dx dx
x x
f dx f dx f dx f dxdx dx
x x
f dx f dx dx f dx
1 2( , )f x x
Extreme values and multivariate functions
Sufficient condition for a local maximum (minimum)
• If the second total derivative evaluated at a
stationary point of a function f(x1,x2) is negative
(positive) for any dx1 and dx2, then that stationary
point represents a local maximum (minimum) of the
function
Extreme values and multivariate functions
Sufficient Condition for a Local Minimum:
Sufficient Condition for a Local Minimum:
22 12
11 22
11
( )0 if 0 and 0
fd y f f
f
2 2
11 11 22 120 if 0 and d y f f f f
Extreme values and multivariate functions
Sufficient Condition for a Local Maximum:
Sufficient Condition for a Local Maximum:
22 12
11 22
11
( )0 if 0 and 0
fd y f f
f
2 2
11 11 22 120 if 0 and d y f f f f
Extreme values of
multivariate functions – bivariate case
• Choose (x1,x2) to maximize (or to minimize) f(x1,x2)
First Order Conditions:
f1 (x1,x2)=0 and f2 (x1,x2)=0
stationary points
* *
1 2( , )x x
Second Order Conditions
Local Minimum if
___________________
Local Maximum if
* *
11 1 2
2* * * * * *
11 1 2 22 1 2 12 1 2
( , ) 0
( , ) ( , ) ( , )
f x x
and
f x x f x x f x x
* *
11 1 2
2* * * * * *
11 1 2 22 1 2 12 1 2
( , ) 0
( , ) ( , ) ( , )
f x x
and
f x x f x x f x x
Exercises
• Choose (x1,x2) to minimize
2 2
1 2 1 2 1 2( , ) 4 2f x x x x x x
2 2
1 2 1 2 1 2
*
1 1 1
*
2 2 2
( , ) 4 2
:
4 2 0 2
14 1 0
4
f x x x x x x
FOC
f x x
f x x
1 1
2 2
11 12 22
2
11 11 22 12
4 2
4 1
: We need to find , ,
If 0 and . ( ) , then local min
f x
f x
SOC f f f
f f f f
1 1
2 2
11
12
22
11
2
11 22 12
4 2
4 1
:
2
0
4
Observe that 2 0 and
. 2(4) 8 0 ( )
1Hence, ( 2, ) is local minimum.
4
f x
f x
SOC
f
f
f
f
f f f
Exercise 2
• Find the local max and local min of
2 2
1 2 1 2 1 2( , ) 8 7 14f x x x x x x
2 2
1 2 1 2 1 2
*
1 1 1
*
2 2 2
( , ) 8 7 14
:
8 2 0 4
14 14 0 1
f x x x x x x
FOC
f x x
f x x
1 1
2 2
11
12
22
11
2
11 22 12
8 2
14 14
:
2
0
14
Observe that 2 0 and
. ( 2)( 14) 28 0 ( )
Hence, (4,1) is local max.
f x
f x
SOC
f
f
f
f
f f f
Exercise 3
• Find the local max and local min of
2 2
1 2 1 2 1 2 1 2( , ) 2 4 16f x x x x x x x x
2 2
1 2 1 2 1 2 1 2
1 1 2
2 2 1
( , ) 2 4 16
:
2 2 0
8 16 0
f x x x x x x x x
FOC
f x x
f x x
1 1 2
2 2 1
1 2 2 1
*
1 1 1
*
2
2 2 0
8 16 0
2 2 0 2 2
8(2 2 ) 16 0 0
2
f x x
f x x
x x x x
x x x
x
1 1 2
2 2 1
11
12
22
11
2
11 22 12
2 2
8 16
:
2
1
8
Observe that 2 0 and
. (2)(8) 16 1 ( )
Hence, (0, 2) is local min.
f x x
f x x
SOC
f
f
f
f
f f f
Exercise 4
• Find the local max and local min of
2 2
1 2 1 2 1 2 1 2
1 1( , )
8 2f x x x x x x x x
2 2
1 2 1 2 1 2 1 2
1 1 2
2 2 1
1 1( , )
8 2
:
1 0
11 0
4
f x x x x x x x x
FOC
f x x
f x x
1 1 2
2 2 1
1 2 2 1
1 1 1 1
*
1
*
2
1 0
11 0
4
1 0 1
1(1 ) 1 0 1 4 4 0
4
1
0
f x x
f x x
x x x x
x x x x
x
x
1 1 2
2 2 1
11
12
22
11
2
11 22 12
1
11
4
:
1
1
1
4
Observe that 1 0 and
1 1. ( 1)( ) 1 ( )
4 4
Hence, no concl.
f x x
f x x
SOC
f
f
f
f
f f f
Exercise 6
• Find the local max and local min of
2 3
1 2 2 1 2
1 1( , )
2 3f x x x x x
2 3
1 2 2 1 2
2 *
1 1 1
*
2 2 2
1 1( , )
2 3
:
0 0
1 0 1
f x x x x x
FOC
f x x
f x x
2
1 1
2 2
11 1
12
22
1
:
2
0
1
f x
f x
SOC
f x
f
f
11 1
12
22
11
2
11 22 12
At (0,1)
:
2 0
0
1
Observe that 0 and
. (0)( 1) 0 0 ( )
Hence, no conl.
SOC
f x
f
f
f
f f f
Exercise 7
• Find the local max and local min of
2 3
1 2 1 2 1 2
1 1( , )
2 3f x x x x x x
2 3
1 2 1 2 1 2
2 * *
1 1 1 1
*
2 2 2
1 1( , )
2 3
:
1 0 1 or 1
1 0 1
Two stationary points (-1,1) and (1,1)
f x x x x x x
FOC
f x x x
f x x
2
1 1
2 2
11 1
12
22
1
1
:
2
0
1
f x
f x
SOC
f x
f
f
11 1
12
22
11
2
11 22 12
At (1,1)
:
2 2
0
1
Observe that 2 0 and
. ( 2)( 1) 2 0 ( )
Hence, (1,1) is local max.
SOC
f x
f
f
f
f f f
11 1
12
22
11
2
11 22 12
At ( 1,1)
:
2 2( 1) 2
0
1
Observe that 2 0 and
. (2)( 1) 2 0 ( )
Hence, at (-1,1) no concl.
SOC
f x
f
f
f
f f f
