, , .
, f (,b) ,
f (a1 1 + a2 2) £ a1 f (1) + a2 f (2),
1 2 (,b); a1, a2 ³ 0, a1 + a2 = 1.
a1 ¹ 0, a2 ¹ 0
f (a1 1 + a2 2) < a1 f (1) + a2 f (2),
f (;b). , (. .)
, , f .
.
4. (, y) 1, y . y = yo Î[0;1] 2,
V = M (x, y),
yo
M (x, yo) = V.
. 4 (, y) y, , , 2 .
. -, .. , , .
, (, y) y, (1), 2 , (2).
1: (, y) y, 1 .
V = M(x,y),
o 1
M (xo, y) = V.
. [0;1]
(, y) = .
x Î[0; 1], y Î(0;1),
(, y) y Î(0;1). , (3)
V = .
, 0 £ £
=
0,5 < £ 1
=
V = max [ ; ] =
= max [ ; ] =
= max [ ; ] = .
o = .
= ,
.. y = 0,
= ,
, = .
, (5) y, , , y Î[0;1]. , .. 2 yo, (4)
|
|
=
, = ,
= .
yo = 0. , 2 yo = 0.
. 1, 2 - , (, y).
, . (, y), , y. 1 . , Î[0; 1], y Î[0; 1] (, y) y, y = 0 y = 1 (, y) = . yo 2 ( 4).
2 1 ,
(, yo) = V.
.
5. y (, y), yo 2 V, :
1) yo = 1, 1 1,
(1, 1) £ 1;
2) yo = 0, 1 2,
(2, 0) ³ 0;
3) 0 £ yo £ 1, 1 , 1 2.
(1, yo) £ 0, (2, yo) ³ 0,
1 a, 2 (1 - a), a
a (1, yo) + (1 - a) (2, yo) = 0.
.
(, y) = ( - y)2 = 2 - 2y + y2.
y, .
= 2 > 0.
, (, y) y, 4 (1), 2 yo, (2). ,
V = (x - y)2;
(x2 - 2xy + y2)
= 2x - 2 y:= 0 Þ x = y
= 2 > 0 Þ x = y M y.
Þ x = 0 () x = 1
M (0; y) = y2
M (1; y) = 1 - 2 y + y2 = (y - 1)2
Þ V= max { y2; (1 - y)2}
max {...} , y2 = (1 - y)2, .. y = .
V = yo = .
1. yo = , 0 < yo < 1. 5 .
(, yo) = V,
( - )2 = .
, 1 = 0, 2 = 1. a 1 = 0. (*).
|
|
a (0, ) + (1 - a) (1, ) = 0.
a :
a - (1 - a) = 0,
a = . , 1
F () = Jo () + J1 (),
2
Q (y) = (y).
(x)
(x) = .