Minima, Maxima, Saddle points
Levent Kandiller
Industrial Engineering Department
Çankaya University, Turkey
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.1/9
Scalar Functions
Let us remember the properties for maxima, minima and
saddle points when we have scalar functions with two
variables with the help of the following examples.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.2/9
Scalar Functions
Example. Let f (x, y) = x2 + y 2 . Find the extreme points:
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.2/9
Scalar Functions
Example. Let f (x, y) = x2 + y 2 . Find the extreme points:
2
1.5
1
0.5
0
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
c
−1
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.2/9
Scalar Functions
Example. Let f (x, y) = x2 + y 2 . Find the extreme points:
2
1.5
1
0.5
0
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
∂f (x,y)
∂x
.
= 2x = 0 ⇒ x = 0,
∂f (x,y)
∂y
.
= 2y = 0 ⇒ y = 0.
Since we have only one critical point, it is either the maximum or the
minimum. We observe that f (x, y) takes only nonnegative values.
Thus, we see that the origin is the minimum point.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.2/9
Scalar Functions
Example. Find the extreme points of
f (x, y) = xy − x2 − y 2 − 2x − 2y + 4.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.3/9
Scalar Functions
Example. Find the extreme points of
f (x, y) = xy − x2 − y 2 − 2x − 2y + 4.
The function is differentiable and has no boundary points.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.3/9
Scalar Functions
Example. Find the extreme points of
f (x, y) = xy − x2 − y 2 − 2x − 2y + 4.
fx =
Thus, x
∂f (x,y)
∂x
= y − 2x − 2, fy =
∂f (x,y)
∂y
= x − 2y − 2.
= y = −2 is the critical point.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.3/9
Scalar Functions
Example. Find the extreme points of
f (x, y) = xy − x2 − y 2 − 2x − 2y + 4.
fx =
Thus, x
fxx =
∂f (x,y)
∂x
= y − 2x − 2, fy =
∂f (x,y)
∂y
= x − 2y − 2.
= y = −2 is the critical point.
∂ 2 f (x,y)
∂x2
c
= −2 =
∂ 2 f (x,y)
∂y 2
= fyy , fxy =
∂ 2 f (x,y)
∂x∂y
= 1.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.3/9
Scalar Functions
Example. Find the extreme points of
f (x, y) = xy − x2 − y 2 − 2x − 2y + 4.
fx =
Thus, x
fxx =
∂f (x,y)
∂x
= y − 2x − 2, fy =
∂f (x,y)
∂y
= x − 2y − 2.
= y = −2 is the critical point.
∂ 2 f (x,y)
∂x2
= −2 =
∂ 2 f (x,y)
∂y 2
= fyy , fxy =
The discriminant (Jacobian) of f at (a, b)
Since fxx
fxx
fxy
fxy
fyy
∂ 2 f (x,y)
∂x∂y
= 1.
= (−2, −2) is
2
= fxx fyy − fxy
= 4 − 1 = 3.
2
< 0, fxx fyy − fxy
> 0 ⇒ f has a local maximum at (−2, −2).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.3/9
Scalar Functions
Theorem. The extreme values for f (x, y) can occur only at
i. Boundary points of the domain of f .
ii. Critical points (interior points where fx
= fy = 0, or points where
fx or fy fails to exist).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.4/9
Scalar Functions
Theorem. If the first and second order partial derivatives of f are
continuous throughout an open region containing a point (a, b) and
fx (a, b) = fy (a, b) = 0, you may be able to classify (a, b) with the
second derivative test:
i.
2
fxx < 0, fxx fyy − fxy
> 0 at (a, b) ⇒ local maximum;
ii.
2
> 0 at (a, b) ⇒ local minimum;
fxx > 0, fxx fyy − fxy
iii.
2
< 0 at (a, b) ⇒ saddle point;
fxx fyy − fxy
iv.
2
= 0 at (a, b) ⇒ test is inconclusive (f is singular).
fxx fyy − fxy
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.4/9
Quadratic forms
Definition. The quadratic term
f (x, y) = ax2 + 2bxy + cy 2
is positive definite (negative definite) if and only if a
(a < 0) and ac − b2 > 0.
c
>0
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.5/9
Quadratic forms
Definition. The quadratic term
f (x, y) = ax2 + 2bxy + cy 2
is positive definite (negative definite) if and only if a > 0
(a < 0) and ac − b2 > 0.
f has a minimum (maximum) at
x = y = 0 if and only if fxx (0, 0) > 0 (fxx (0, 0) < 0)
2
and fxx (0, 0)fyy (0, 0) > fxy
(0, 0).
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.5/9
Quadratic forms
Definition. The quadratic term
f (x, y) = ax2 + 2bxy + cy 2
is positive definite (negative definite) if and only if a > 0
(a < 0) and ac − b2 > 0.
f has a minimum (maximum) at
x = y = 0 if and only if fxx (0, 0) > 0 (fxx (0, 0) < 0)
2
and fxx (0, 0)fyy (0, 0) > fxy
(0, 0).
If f (0, 0) = 0, we term f as positive (negative)
semi-definite provided the above conditions hold.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.5/9
Quadratic forms
Now, we are able to introduce matrices to the quadratic
forms:

 
2
2
ax + 2bxy + cy = [x, y] 
c
a b
b c
x

y
.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.6/9
Quadratic forms
Thus, for any symmetric A, the product f = xT Ax is a pure
quadratic form: it has a stationary point at the origin and no
higher terms.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.6/9
Quadratic forms
Thus, for any symmetric A, the product f = xT Ax is a pure
quadratic form: it has a stationary point at the origin and no
higher terms.



x1
a11 a12 · · · a1n



 a21 a22 · · · a2n   x2 



xAT x = [x1 , x2 , · · · , xn ]  .
 . 
.
.
.
 ..
..   .. 
..
..



xn
an1 an2 · · · ann
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.6/9
Quadratic forms
Thus, for any symmetric A, the product f = xT Ax is a pure
quadratic form: it has a stationary point at the origin and no
higher terms.



x1
a11 a12 · · · a1n



 a21 a22 · · · a2n   x2 



xAT x = [x1 , x2 , · · · , xn ]  .
 . 
.
.
.
 ..
..   .. 
..
..



xn
an1 an2 · · · ann
Pn Pn
2
2
= a11 x1 + a12 x1 x2 + · · · + ann xn = i=1 j=1 aij xi xj .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.6/9
Quadratic forms
Definition. If A is such that aij
=
∂2f
∂xi ∂xj (hence
symmetric), it is called the Hessian matrix.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.7/9
Quadratic forms
Definition. If A is such that aij
=
∂2f
∂xi ∂xj (hence
symmetric), it is called the Hessian matrix.
If A is positive definite (xT Ax > 0, ∀x 6= θ ) and if f has
a stationary point at the origin (all first derivatives at the
origin are zero), then f has a minimum.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.7/9
Quadratic forms
Remark. Let f : Rn 7→ R and x∗ ∈ Rn be the local minimum,
∇f (x∗ ) = θ and ∇2 f (x∗ ) is positive definite.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.8/9
Quadratic forms
Remark. Let f : Rn 7→ R and x∗ ∈ Rn be the local minimum,
∇f (x∗ ) = θ and ∇2 f (x∗ ) is positive definite. We are able to explore
the neighborhood of x∗ by means of x∗ + ∆x, where k∆xk is
sufficiently small (such that the second order Taylor’s approximation is
pretty good) and positive.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.8/9
Quadratic forms
Remark. Let f : Rn 7→ R and x∗ ∈ Rn be the local minimum,
∇f (x∗ ) = θ and ∇2 f (x∗ ) is positive definite. We are able to explore
the neighborhood of x∗ by means of x∗ + ∆x, where k∆xk is
sufficiently small (such that the second order Taylor’s approximation is
pretty good) and positive. Then,
f (x∗ + ∆x) ∼
= f (x∗ ) + ∆xT ∇f (x∗ ) + 21 ∆xT ∇2 f (x∗ )∆x.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.8/9
Quadratic forms
Remark. Let f : Rn 7→ R and x∗ ∈ Rn be the local minimum,
∇f (x∗ ) = θ and ∇2 f (x∗ ) is positive definite. We are able to explore
the neighborhood of x∗ by means of x∗ + ∆x, where k∆xk is
sufficiently small (such that the second order Taylor’s approximation is
pretty good) and positive. Then,
f (x∗ + ∆x) ∼
= f (x∗ ) + ∆xT ∇f (x∗ ) + 21 ∆xT ∇2 f (x∗ )∆x.
The second term is zero since x∗ is a critical point and the third term is
positive since the Hessian evaluated at x∗ is positive definite. Thus, the
left hand side is always strictly greater than the right hand side,
indicating the local minimality of x∗ .
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.8/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
Let f (x1 , x2 ) = 13 x31 + 12 x21 + 2x1 x2 + 12 x22 − x2 + 19. Find the stationary
and boundary points, then find the minimizer and the maximizer over
−4 ≤ x2 ≤ 0 ≤ x1 ≤ 3
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
Let f (x1 , x2 ) = 13 x31 + 12 x21 + 2x1 x2 + 12 x22 − x2 + 19. Find the stationary
and boundary points, then find the minimizer and the maximizer over
−4 ≤ x2 ≤ 0 ≤ x1 ≤ 3
E
33
32
31
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
2,2
2
1,6
-3,5
1,8
1,2
1,4
1
-2,9
0,8
0,4
0,6
0
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
-4
3
2,8
2,6
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
2,4
c
0,2
19
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
Let f (x1 , x2 ) = 13 x31 + 12 x21 + 2x1 x2 + 12 x22 − x2 + 19. Find the stationary
and boundary points, then find the minimizer and the maximizer over
−4 ≤ x
2 ≤ 0 ≤
 x1 ≤ 3

  
∂f
 (x − 1)(x − 2) = 0
2
1
1
 ∂x1   x1 + x1 + 2x2  .  0 
∇f (x) = 
⇒
=
=

∂f
x2 = 1 − 2x1
2x1 + x2 − 1
0
∂x2
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4

∇f (x) = 
∂f
∂x1
∂f
∂x2
3
2,8
2,4
2,6
2
2,2
1,4
1,6
1
1,2
0,4
0,8
-3,5
-4

  

 (x − 1)(x − 2) = 0
2
1
1
  x1 + x1 + 2x2  .  0 
=
⇒
=


x2 = 1 − 2x1
2x1 + x2 − 1
0
2
Therefore, xA = 4
1
−1
3
5,
2
xB = 4
defined by −4 ≤ x2 ≤ 0 ≤ x1 ≤ 3.
c
-2,9
1,8

0,6
0

0,2
19
2
−3
3
5 are stationary points inside the region
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
∂f
∂x1

∇f (x) = 
∂f
∂x2
c
3
2,8
2,4
2,6
2
2,2
1,8
1,4
1,6
1
1,2
0,4
0,8
-3,5
-4

  

 (x − 1)(x − 2) = 0
2
1
1
  x1 + x1 + 2x2  .  0 
=
⇒
=


x2 = 1 − 2x1
2x1 + x2 − 1
0
1
−1
3
5,
3
5,
2
xB = 4
2
xII = 4
defined by −4 ≤ x2 ≤ 0 ≤ x1 ≤ 3.
defined by
-2,9

2
Therefore, xA = 4
2
xI = 4
0,6
0

0,2
19
0
x2
2
xC = 4
0
0
3
5,
3
x2
2
xD = 4
2
−3
3
5 are stationary points inside the region
3Moreover, we 2have the3 following boundaries
2
3
5 and xIII = 4 x1 5 , xIV = 4 x1 5
0
−4
3
5,
2
xE = 4
0
−4
3
0
3
5,
xF
2
=4
3
−4
3
5.
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
∇2 f (x)

=
c
∂2f
∂2f
∂x1 ∂x1
∂x1 ∂x2
∂2f
∂x2 ∂x1
∂2f
∂x2 ∂x2


-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
Let the Hessian
matrix be

0,2
19

  2x1 + 1 2 
.
=
2
1
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
∂2f
∂2f
∂x1 ∂x1
∂x1 ∂x2


-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
Let the Hessian
matrix be

0,2
19

  2x1 + 1 2 
. Then, we
=
2
2
∂ f
∂ f
2
1
∂x2 ∂x1
∂x2 ∂x2




3 2
5 2
2
2



.
have ∇ f (xA ) =
and ∇ f (xB ) =
2 1
2 1
∇2 f (x)

=
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
∂2f
∂2f
∂x1 ∂x1
∂x1 ∂x2


-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
Let the Hessian
matrix be

0,2
19

  2x1 + 1 2 
. Then, we
=
2
2
∂ f
∂ f
2
1
∂x2 ∂x1
∂x2 ∂x2




3 2
5 2
2
2



.
have ∇ f (xA ) =
and ∇ f (xB ) =
2 1
2 1
∇2 f (x)

=
Let us check the positive definiteness of ∇2 f (xA ):
2
v T ∇2 f (xA )v = [v1 , v2 ] 4
3
2
2
1
32
54
v1
v2
3
5 = 3v12 + 4v1 v2 + v22 .
If v1 = −0.5 and v2 = 1.0, we will have v T ∇2 f (xA )v < 0. On the other hand, if
v1 = 1.5 and v2 = 1.0, we will have v T ∇2 f (xA )v > 0. Thus, ∇2 f (xA ) is indefinite.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
∂2f
∂2f
∂x1 ∂x1
∂x1 ∂x2


-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
Let the Hessian
matrix be

0,2
19

  2x1 + 1 2 
. Then, we
=
2
2
∂ f
∂ f
2
1
∂x2 ∂x1
∂x2 ∂x2




3 2
5 2
2
2



.
have ∇ f (xA ) =
and ∇ f (xB ) =
2 1
2 1
∇2 f (x)

=
Let us check ∇2 f (xB ):
2
v T ∇2 f (xB )v = [v1 , v2 ] 4
32
54
3
5 2
v1
5 = 5v12 + 4v1 v2 + v22 = v12 + (2v1 + v2 )2 > 0.
2 1
v2
2
3
2
5 is a local minimizer with
Thus, ∇2 f (xB ) is positive definite and xB = 4
−3
f (xB ) = 19.166667
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
c
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
Let us check the boundary defined by xI :
1 2
df (0, x2 )
.
f (0, x2 ) = x2 − x2 + 19 ⇒
= x2 − 1 = 0 ⇒ x2 = 1.
2
dx2
Since
d2 f (0,x2 )
dx22
= 1 > 0, x2 = 1 > 0 is the local minimizer outside
the feasible region. As the first derivative is negative for
−4 ≤ x2 ≤ 0, we will check x2 = 0 for minimizer and x2 = −4 for
maximizer.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
Let us check the boundary defined by xII :
1 2
65
df (3, x2 )
.
f (3, x2 ) = x2 + 5x2 +
= x2 + 5 = 0 ⇒ x2 = −5.
⇒
2
2
dx2
Since
d2 f (0,x2 )
dx22
= 1 > 0, x2 = −5 < −4 is the local minimizer
outside the feasible region. As the first derivative is positive for
−4 ≤ x2 ≤ 0, we will check x2 = −4 for minimizer and x2 = 0 for
maximizer.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
Let us check the boundary defined by xIII :
1 3 1 2
df (x1 , 0)
.
f (x1 , 0) = x1 + x1 +19 ⇒
= x21 +x1 = 0 ⇒ x1 = 0, −1.
3
2
dx1
d2 f (x1 ,0)
Since dx2 = 2x1 + 1, x1 = 0 is the local minimizer
1
2
= 1 > 0) on the boundary, and x1 = −1 is the local
( d fdx(0,0)
2
1
d2 f (−1,0)
maximizer ( dx2
= −1 < 0) outside the feasible region. As
1
the first derivative is positive for 0 ≤ x2 ≤ 3, we will check x2 = 3
for maximizer.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
Let us check the boundary defined by xIV :
f (x1 , −4) = 13 x31 + 12 x21 − 8x1 + 31 ⇒
df (x1 ,−4)
dx1
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
.
= x21 + x1 − 8 = 0
√
d2 f (x1 ,−4)
−1± 1+32
. Since
= 2x1 + 1 again, the positive
⇒ x1 =
2
dx21
√
root x1 = −1+2 33 = 2.3723 is the local minimizer
d2 f (2.3723,0)
> 0), and the negative root is the local maximizer but
(
dx21
it is outside the feasible region. As the first derivative is positive
for 0 ≤ x2 ≤ 3, we will check x2 = 3 for maximizer again.
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9
E
33
32
31
Collaborative Work:
30
29
C
28
27
A
26
25
24
23
-0.4
-0.9
22
B
21
-1,4
-1,9
20
-2,4
-4
3
2,8
2,4
2,6
2
2,2
-3,5
1,8
1,4
1,6
1
1,2
-2,9
0,8
0,4
0,6
0
0,2
19
To sum up, we have to consider (2, −3), (0, 0) and (2.3723, −4)
for the minimizer; (3, 0) and (0, −4) for the maximizer:
f (2, −3) = 19.16667, f (0, 0) = 19, f (2.3723, −4) = 19.28529
⇒ (0, 0) is the minimizer!
f (3, 0) = 32.5, f (0, −4) = 31 ⇒ (3, 0) is the maximizer!
c
Levent Kandiller, Principles of Mathematics in Operations Research, The International Series in
Operations Research and Management Science, Vol. 97, Springer, 2007. ISBN: 0-387-37734-4
Minima, Maxima, Saddle points – p.9/9