Cho A ld tdp con kh6c rSng trong kh6ng gian metric X, d.. Hdy ki6m tra sy x6c clinh cna f Q vd chrmg minh f tuy}ntinh li€n tpc.. Ghi chti: Cdn bQ coi thi khong giai thich gi th€m.
Trang 1sg cilo DUc vA DAo rAo
D4.I HQC HUE
Ho vd tEn thl sinh:
56 bdo danh.
ri'rm rUYEN SINH sAU DAI HQC NAM 2012 ( Egt 1)
Mdn thi: GIAI TiCH
@anh cho cao hqc) Thdi gian lam bdi: 180 Phut
CAu 1
a) Khao s6t tinh kha vi cua hdm
, \ ( 9'n6u (x; y) +(o; o)
f ( x , ! ) = l * , * y ' \ ' r
[o n6u (x; !) = (0; o)
b) Tim mi0n hQi tu cua chu6i nam lfly thua
@
Y ( - 2 ) " ( x _ 1 , ) , .
L n * L c) Tinh tich phdn ducrng
I G.siny + 2xy)d.x * (x' + e' cosY)dY
|
trong d,6 L lii cung cua parab ol x - y2 chAy tu di6m 0 (0;0) d6n A(L; 1)
Ciu 2 Cho A ld tdp con kh6c rSng trong kh6ng gian metric (X, d) Chrmg minh ring, him sO;, X + IR x6c dinh bdi
f (r) = d(x; A) : IEId(a; x)
liOn tuc trOn X vd tap hap M -
{* X: 0 s (d(*; A))' + d.(x; A) s 2} dong trong X
) -Ciu 3 Xdt tAp X g6m c6c hdm thgc x = x(t) liOn tpc tr0n [0; +*) sao cho
"i'Tl*'
a) Chtmg minh (X; ll ll) h khdng gian dinh chuAn voi
l l r f l - s u p e t l x ( t ) l , v x e X
r e [o ; + m ) b) Xet phi6m ham f , X + IR, sao cho f (x) - Io** tx(t) dt Hdy ki6m tra sy x6c clinh cna f Q) vd chrmg minh f tuy}ntinh li€n tpc Tinh ll/ll
Cffu 4
a) Chokh6ng gianHilbert H vitM ldt4p contrum4ttrong cuaH Gid su x e H vit (x,")n ld ddy bi ch4n trong H sao cho v6i m6i y e M thi limrl-+@(xn;!l = (x;yl Chimg minh ring ddy (x) r, hQi tq y6u d€n x
b) Tr€n mQt khdng gian Hilbert H, vsimoi a * 0 chrmg minh ring, phi6m ham tuy6n rinh li€n t.uc fr(x) = (x; al ldmQt todn 6nh vd suy ra fo cingld mOt 6nh x4 mo.
Ghi chti: Cdn bQ coi thi khong giai thich gi th€m.