Saturday, November 1, 2014

The Function x^2 + xy + y^2 – Part 6

6)      Every other row, column and diagonal of x2 + xy + y2 has a minimum value, which all work together to form lines passing through 0.

Every other row, column and diagonal of x2 + xy + y2 has a minimum value, which all work together to form lines passing through 0.

-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10

10
100
91
84
79
76
75
76
79
84
91
100
111
124
139
156
175
196
219
244
271
300
10
9
91
81
73
67
63
61
61
63
67
73
81
91
103
117
133
151
171
193
217
243
271
9
8
84
73
64
57
52
49
48
49
52
57
64
73
84
97
112
129
148
169
192
217
244
8
7
79
67
57
49
43
39
37
37
39
43
49
57
67
79
93
109
127
147
169
193
219
7
6
76
63
52
43
36
31
28
27
28
31
36
43
52
63
76
91
108
127
148
171
196
6
5
75
61
49
39
31
25
21
19
19
21
25
31
39
49
61
75
91
109
129
151
175
5
4
76
61
48
37
28
21
16
13
12
13
16
21
28
37
48
61
76
93
112
133
156
4
3
79
63
49
37
27
19
13
9
7
7
9
13
19
27
37
49
63
79
97
117
139
3
2
84
67
52
39
28
19
12
7
4
3
4
7
12
19
28
39
52
67
84
103
124
2
1
91
73
57
43
31
21
13
7
3
1
1
3
7
13
21
31
43
57
73
91
111
1
0
100
81
64
49
36
25
16
9
4
1
0
1
4
9
16
25
36
49
64
81
100
0
-1
111
91
73
57
43
31
21
13
7
3
1
1
3
7
13
21
31
43
57
73
91
-1
-2
124
103
84
67
52
39
28
19
12
7
4
3
4
7
12
19
28
39
52
67
84
-2
-3
139
117
97
79
63
49
37
27
19
13
9
7
7
9
13
19
27
37
49
63
79
-3
-4
156
133
112
93
76
61
48
37
28
21
16
13
12
13
16
21
28
37
48
61
76
-4
-5
175
151
129
109
91
75
61
49
39
31
25
21
19
19
21
25
31
39
49
61
75
-5
-6
196
171
148
127
108
91
76
63
52
43
36
31
28
27
28
31
36
43
52
63
76
-6
-7
219
193
169
147
127
109
93
79
67
57
49
43
39
37
37
39
43
49
57
67
79
-7
-8
244
217
192
169
148
129
112
97
84
73
64
57
52
49
48
49
52
57
64
73
84
-8
-9
271
243
217
193
171
151
133
117
103
91
81
73
67
63
61
61
63
67
73
81
91
-9
-10
300
271
244
219
196
175
156
139
124
111
100
91
84
79
76
75
76
79
84
91
100
-10

-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10


On even rows, where y = 2n, so x2 + xy + y2 = x2 + 2nx + 4n2.  The minimum would be the vertex x = -b/(2a) = -2n/(2∙1) = -n, so each minimum goes through (–n, 2n) which can be represented by y = -2x, a line going through (0, 0).  A similar argument can be made for even columns, but switching x and y so that each minimum goes through (2n, -n) which can be represented by y = -½x, which is also a line going through (0, 0).  On even diagonals of negative slope, x + y = 2n, so y = -x + 2n, so x2 + xy + y2 = x2 + x(-x + 2n) + (-x + 2n)2 = x2 – x2 + 2nx + x2 – 4nx + 4n2 = x2 – 2nx + 4n2.  The minimum would be the vertex x = -b/(2a) = -(-2n)/(2∙1) = n, so y = -x + 2n = -n + 2n = n, so each minimum goes through (n, n) which can be represented by y = x, yet another line going through (0, 0).  Finally, on even diagonals of positive slope, y – x = 2n, so y = x + 2n, so x2 + xy + y2 = x2 + x(x + 2n) + (x + 2n)2 = x2 + x2 + 2nx + x2 + 4nx + 4n2 = 3x2 + 6nx + 4n2.  The minimum would be the vertex x = -b/(2a) = -(6n)/(2∙3) = -n, so y = x + 2n = -n + 2n = n, so each minimum goes through (-n, n) which can be represented by y = -x, yet another line going through (0, 0).

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