skkn một số bài toán cực trị trong hình học giải tích lớp 12 thpt trần phú
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( )
( ) ( )
t
t q
− −
′
= +
− + − +
f t
( )
( ) ( )
t
` `
t q
− −
′
= ⇔ + =
− + − +
f t
( ) ( )
b td b d
t q
− − −
⇔ =
− + − +
b•d
"'234€€t+8!
b•d
( ) ( ) ( ) ( )
t y ~ y t q~ ⇔ − − + = − − +
( ) ( )
t b d
t q
t b d
− = −
⇔ − = − ⇔
− = − −
ypt~
e
t
= ∉
⇔
=
O#/+-$:>#&'!
−∞
e
t
+∞
#%&' U`n
#&'
+∞
+∞
tx `
t
+
Gia?@AB8CD"89>5E5 F> GH1 Trang
13/33
Y#:1+
( )
e
$
t
# #
=
÷
o
tx `
t
+
jI1hmnhO'.@=G
tx `
t
+
''?
e
t
=
;-
hb
e
t
p`p`d
%N%
6(&8+$:>
(
)
*
= +
= +
= +
+'0$Abppd8"
bppd=
r
u
"-
bptp d= −
uuur
AB
+8
r
u
*
uuur
CD
oqn_Qoa•`
⇒
3<"<8"mO"-
y ~
r uuur uuur
u AB NA
obUupqpdbUpUp`do`Qqo_
⇒
"-mOD+
UX"+$hmO-obhmnhOnmOdmO3<'P
'.@=3hmnhO'.@=*
zD'0$
b p p d ∈ ⇒ + +M d M 2+2
+$'0hmnhO'.
@=
zD
( )
hmnhOo b d b d b d b d b d#
= + + + + + + − + +
( )
o a u a _ u#
+ + + − +
o
bt d bt d q
+ + + − +
X8'-$
t t
{b d
bt d bt d q
#
+ −
= +
+ + − +
t t
{b d ` `
bt d bt d q
#
+ −
= ⇔ + =
+ + − +
t bt d
bt d bt d q
+ − −
⇔ =
+ + − +
"
t t
− ≤ ≤
bt d ybt d q~ bt d ybt d ~
⇔ + − + = − + +
Gia?@AB8CD"89>5E5 F> GH1 Trang
14/33
b$S[IBX'(&
( )
!
x-2 y-2 z -1
d = =
1
"-+'0$mbUppdObp
qp`d*J1$'0$h:+"+$hmO'.@
=
u
bt d t
t
qbt d bt d
bt d t
t
t
−
=
+ = −
⇔ + = − ⇔ ⇔ ⇔ = −
+ = − +
= −
O#/+-$:>#&'!
−∞
t
−
+∞
#%&' U`n
#&'
+∞
+∞
t
+=1#&''.@=G
t
3o
t
−
+1"
q
p p d
t t t
M(
hmnhO'@=G
t
"Y#Z[2J8#*HR#*28K##EI#&'2Hc#*3"F]#*3"K3>"N8<K2"E!<=
2" (%a$2 !2<d-]#*%N#"]#
aeB"8"%-FG#*2"4#*H
:H
$"[8#"' !$K$-%.!
∈
H
:
∈
H
/E$"`#-8R#(')#**+$$"'#*$,"%-FG#*2J0#
G%*%N%
U j# + '( &
+$:>
U ‚=1 h
∈
d
"- A
∈
d
b + '
+$:>d*
U M 4
* `
=
uuuur r
MN u
"-
* `
=
uuuur r
MN u
b
r
u
r
u
-"D9
/+
"-
d*
U $+' hA"-3#I*
Gia?@AB8CD"89>5E5 F> GH1 Trang
15/33
b$S[(!X+'(&
!
x-5 y+1 z -11
= =
-1
!
e t
−
x+ 4 y-3 z - 4
= =
d X;$
D+
d $'0$h
∈
"-A
∈
:+' -hA\=*
%N%
d
+h
bupUpd8"
bpp d= −
uur
u
+h
bUqptpqd8"
b epptd= −
uur
u
+8y
uur
u
uur
u
~
uuuuuur
M M
obxpqp_dbUapqpUedoUen_QoU_x
`
≠
+1
"-
D+*
d*h
∈
"-A
∈
:+' -hA\=3"-93hA-'
-'"<8/+
"-
*
R+$:>/++'(&
!
u
= +
= − +
= −
x t
y
z
!
q e
t
q t
= − −
= +
= +
x t
y
z
h
∈
hbunpUnpUdA
∈
AbUqQekptnkpqntkd
b
=
uuuur
MN
UekUQapkQnqptknQed
+8
* ` _ { _ _ `
_ { _ u` ` {
* `
= − − + = =
⇔ ⇔
+ + = = −
=
uuuur r
uuuur r
MN u
MN u
N'8hbeptpad"-Abtppd
jI1"hbeptpad"-Abtppd' -hA\=G
*
%N%
U ‚=1'0$hM-#"<
8/+hmO
U +$hmO84^o
AB.MH
'
.@=3h@=+1h-
'"<8/+mO"-*
+=1+h
bpqpUd8"
bpp`d=
r
u
mO+mbpptd"-
=
uuur
AB
b`pUpUdo
−
uur
u
Gia?@AB8CD"89>5E5 F> GH1 Trang
16/33
b$S[)BX'(&!
q
= +
= +
= −
x t
y
z
"-+'0$mbpptdObp`pd*$
'0$h:++$hmO84@=
"
b`ppd=
uur
u
-"D9/+mO
R+$:>mO
{
t {
=
= +
= +
x
y
z
hbnpqnpUd
∈
bpnkptnkd
∈
AB
b
=
uuuur
MH
UUpkQUpknud
+8
* `
{ t { t
{ t t
* `
=
− = = −
⇔ ⇔
− = − = −
=
uuuur r
uuuur uur
MH u
MH u
jI1hbUppUdbpUp`d3'8ho
t
mOo
N4
_
,
∆
= =
MAB
AB.MH
%N%
U M:B$%Eb^d87$T3ƒ#lF"h#lF"
|lA
U +=1ƒoThnTA„hA'8$%Eb^d8'(3@=
-ƒohA3"-93hA@=+1hA-'"<8
/+"-|l*
6(&+hb`p`pd8"
b`pp d= −
r
u
|l+|b`p`p`d8"
bp`p`d" =
r
y
r
u
"
r
~
uuuur
OM
ob`p`pUdb`p`pdoU
`
≠
"-|lD+*
jhb`ppUd∈Abkp`p`d∈|l"-
b
=
uuuur
MN
kpUpQd
+8
* ` `
{ ` { `
* `
"
= − − + = =
⇔ ⇔
= =
=
uuuur r
uuuur r
MN u
MN
jI1hb`ppdAb`p`p`d…|
Gia?@AB8CD"89>5E5 F> GH1 Trang
17/33
b$S[IBX'(&!
`
=
=
= −
x
y
z
*$%E#lF
"+'(&"-C|lJ1"#$%Eb^d
83@=*
h%Eb^d87$Tb`
p p d
3ƒo
=
MN
R$%Eb^d!
b d b d
( ) *+ − + − =
)) %A\8WX]@$fWXYg hiVj
' h
a(B "8"%-%.!3"`#^%a29:\%&2
3"F]#*2J #"!123"4#*567-%e'9(E$K$"
!P2>"8N#*/;##"M2
G%*%N%B
-#"<8/+O$%
&bid3'8+$mO"<"-
3bOpbiddoO€mO*jI1bOpbidd
=GmO3m…3'8 bid-$%
&'+m"-"<8"mO*
%N%B
bid'0$TbtpUpUd$ 3=3bid-$%&'+N
"-"<8"NT*
bidI
bp
=
uur
DI 1; -5)
-$"1#
R$%&bid!blUdnb1ndQubwUtdo`
⇔
ln1Quwnuo`
%N%
h%Eb^d83ƒobmpbidd=3bid+O"-"<8
"mO
bp
=
uuur
BA 2; 2)
-"D1#/+bid
Rbid!blUdnb1ndnbwQdo`
⇔
ln1nwQo`
Gia?@AB8CD"89>5E5 F> GH1 Trang
18/33
b$S[(Bj#$%&bid'+'0$NbpUptd"-
'0$TbtpUpUd$ 3=*
b$S[)BX+'0$mbpptdObpUpdbid-$%&+m*
$%E7$m"-#lF"bidJ1"#$%
Eb^d83=*
ƒobmpbidd
_
t
+ + −
=
+ +
R$%Eb^d!blUd
nb1Ud
nbwQtd
oa*
a)B"8-%.!9(E-FG#*2"4#*f>")#*-%e'9\%&23"F]#*
2J #"!123"4#*567$"Wf<8$"8>"8N#*$K$"2g9-ȷ/;##"M2
G%*%N%B
M-#"<8/+m
$%&bidV-#"<8/+m
k
+8bmpbiddom€mV=
…V3'8bid-$%&'+k"-"<
8"mV*+1bid+ k "-"<8"
$b†md*
%N%!
h%&bid'++'0$mO"-X$ 3=3bid
'++'0$mO"-"<8"$bmOXd*
bp p d= − −
uuur
AB
b p tp d= − − −
uuur
AC
bmOXd8"D1#
y ~ b pqp ud= = − −
r uuur uuur
n AB AC
bid8"D1#
y ~ b a _p td tbtppd
α
= = − − − = −
uur r uuur
n n AB
Rbid!tblQdnb1QdnbwQtdo`
⇔
tln1nwQo`
%N%
d
+h
bppUd8"
bpp d= −
uur
u
Gia?@AB8CD"89>5E5 F> GH1 Trang
19/33
b$S[(!X+'0$mbpptdObtp`pdpXb`pUpd*j#
$%&bid'++'0$mO"-X$ 3
=*
b$S[)BX+(&
!
( ) *
− − +
= =
−
t
!
q q
( ) *
− −
= =
− −
d X;$+'(&::"+*
d $%&;+
J1"#$%&bid
:+3]+
"-bid-=*
+h
b`ptpd8"
b p qpqd= − −
uur
u
+=1
= −
uur uur
u u
"-
h
∉
+'(&::"+*
dzDbi
d-$%&;+
"-
bi
d8"D1#
y ~ bxpp_d bqpptd
= = = =
r uur uuuuuur uur
1 1 2
n u M M n
V]+
"-bid-=3bid"<8"bi
d*
N'8bidI
y ~ bxp p ed
= − −
uur uur
1 2
u n
-"D1#+h
bppUd*
Rbid!xblUdUb1UdUebwndo`
+1xlQ1QewQo`*
aIB"8!123"4#*567(E-%.!92"'P$567:/MI>")#*2"'P$567
!-FG#*2"4#*f#b!2J8#*567-%e'9(E$K$"!P2>"8N#*/;#
#"M2:#"?#"M2
G%*%N%
M-#/+Ok+=1bOpklmO€mO
jI13YO'#k=3
m…+1†-'(&G$
bid"-"<8"mO*
MV-#"<8/+
Obid3'8bOpbiddoO„OV
jI13YO'#k@=3
V…+1†-'(&'++
'0$mV*
%N%
+=1bid8"D1#
bp pd
α
= −
uur
n
dM-#"<8/+Obid
RO!
t
u
= +
= −
= +
x t
y
z t
+' '0$;"-4$/+!
bndUbtQdnunnuo`
⇔ = −
+1bUpeptd
Gia?@AB8CD"89>5E5 F> GH1 Trang
20/33
b$S[(!X$%&bid!lQ1nwnuo`"-'0$mbUtptpUtd*
j#'(&†G$bid+'0$m"-'0$
Obptpud$ 3!
dA@=* d‚=*
+=1bOp†d@=3†'++'0$m"I1
bpqp_d=
uuur
AH
-
"D9/+†*
R/+†!
q _
= =
x+3 y-3 z +3
d+=1bOp†d=3†-'(&G$bid+m"-
"<8"mO*
†8"D9
y ~ b_pp `d
α
∆
= = −
uur uuur uur
u AB n
R/+†!
_ `
= =
−
x+3 y-3 z +3
%N%!
zD$%&bid+X"-"<8"bidI
bpp d
=
r
u 3
-$"D
1#†G$bid*
N"I1bNp†d=3†G$bid+X"-"<8"XN*
†8"D9
y ~ bp xpud
α
∆
= = −
uur uuur uur
u CD n
R†!
x u
= =
−
x-2 y+1 z -3
%N%
d6(&+'0$hbp`p`d8"
bp`p d
=
r
u -1
b pp`d= −
uuur
MB
y ~ bppd bppd
α
= = =
uur uuur uur
u MB n
Gia?@AB8CD"89>5E5 F> GH1 Trang
21/33
b$S[)Bj#'(&†'+'0$XbpUptd"<8
"'(&!
t
= =
x-3 y+2 z +5
"-'0$NbqpUpd$ 3
=*
b$S[IBX+'0$mbppUdObUpp`d"-'(&!
`
= +
=
= −
x t
y
z t
d j#$%&bid'+"-O*
d j#'(&†
'+O\:+3
Ym'#†
=*
td j#'(&†
'+O\:+3
Ym'#†
@=*
bid'+OI
bppd
α
=
uur
n
-$"D1#
Rbid!ln1nwQo`
dM-#/+mbid'0bm†
d@=3†
'++
'0$O*
R+$:>m!
= +
= +
= − +
x t
y
z
+' ;"-4$!
nnnUnQo`
t `
t
⇔ + = ⇔ = −
u q
b p p d
t t t
−
⇒
x q q q q
b p p d bp p d
t t t t t
− −
= = − − =
uuur uur
1
BH u
⇒
†
I
uur
1
u
-$"D9
+=1
uur
1
u
"-
r
u
3<g"-†
\+bg
$%&bidd
jI1†
!
= =
− −
x+1 y-2 z
tdMV-#/+m†
+8bm†
domV€mO'0bm†
d
=3V…O+1†
G$bid"-"<8"mO*
+8
y ~ b`p qpqd qb`pp d q
α
= − = − − = −
uur uuur uur
n AB u
⇒
†
I
uur
u
-$ "D 9
$%3
uur
u
"-
r
u
3<g"-†
\+b
g $%&bidd
R†
!
= −
= +
= −
x
y
z
"ST
$+2".Hc#*3"F]#*3"K3>"N8<K2"E!<=-h*%N%T(ETi2J8#*
(jHki
M†
-'(&‡Z'+O"-\:B†
\'0$
Abn`pUd3'8†8"D9
b pp d = − −
uuur
NB
+8
b tppd= −
uuur
AB
y ~ b p pq d = − − −
uuur uuur
NB AB
j-bmp†do
y ~
b d b d bq d
b d b d
− + − + −
=
− − + +
uuur uuur
uuur
NB AB
NB
o
t `
q
− +
+ +
zD-$:>
t `
b d
q
#
− +
=
+ +
8
_ _q
{b d
b qd
#
−
=
+ +
"$
∈
ƒ
Gia?@AB8CD"89>5E5 F> GH1 Trang
22/33
{b d `
#
=
= ⇔
= −
O#/+
b d#
−∞
-2
+∞
#%&' n`U`n
#&'
11 3
t
t
Y#+=1!
• bmp†d=G
3oU
⇒
AbUp`pd
b`pp d b`pp d= − = −
uuur
NB
"-'(&E$8-!
= −
= +
= −
x
y
z
• bmp†d@=G
t
3o
⇒
Abtp`pUd
b qppd bp p d= − = − − −
uuur
NB
"-'(&E$8-!
= =
− −
x+1 y-2 z
acB"8!123"4#*567(E-%.!92"'P$567:-FG#*2"4#*H>")#*
<8#*<8#*"81$#b!2J0#567(E>")#*-%
e'9 !-FG#*2"4#*f#b!2J0#567:
-%e'9<8$"8>"8N#*$K$"*%lf(EH
/E/;##"M2
G%*%N%
M
-'(&+m"-:
:"O-+'0$/+"bid*
zDbRd-$%&b
†d"-T-
#"<8/+ObRd"-
*
+=13]+†"--
O"-O€OTO=3T…3'8†8"
y ~
α
∆
=
uur uur uur
u BI n
*
Gia?@AB8CD"89>5E5 F> GH1 Trang
23/33
%N%
6(&8"
=
r
u
bppUdbid8"
α
=
uur
n
bpUpd
R+$:>!
t
= +
= +
= −
x t
y
z t
MO-+'0$/+"-bid+' O;"-4$!
nQQQtnnqo`
⇔
oU
⇒
Ob`p`pqd
zD
-'(&+m"-::"
R+$:>'(&
!
= − +
= +
= −
x t
y
z t
MT-#"<8/+O
⇒
TbUnpnpQd
=
uur
BI
bUnpnpUuQd
+8
* `
=
uur r
BI u
⇔
UnnbndQbUuQdo`
⇔
oU
⇒
TbUpUpd
6(&†8"
y ~
α
∆
=
uur uur uur
u BI n
obUupU`pqd
R†!
u ` q
= =
− −
x+1 y-1 z -1
%N%
h%&bid+m"-::"bRd8!ln1Qwno`
osG$bid*
6(&†8"
=
r
u
bppUtdbid8"
α
=
uur
n
bppUd
Gia?@AB8CD"89>5E5 F> GH1 Trang
24/33
b$S[(BX'(&!
= =
−
x-1 y-2 z -3
$%&bid!lQ1Qwnq
o`"-'0$mbUppd*j#'(&†G$bid'
+m:+3]+†"--=*
b$S[)BX$%&bRd!ln1Qwno`'0$mbpUpd"-'(
&†
!
t
−
x+1 y z-4
= =
*'(&'+m"-::
:"bRdJ1"#'(&:+3
]+"-†=*
R+$:>†!
q t
= − +
=
= −
x t
y
z t
MO-+'0$/+†"-bid+' O;"-4$!
UnnQbqUtdno`
⇔
o
⇒
Ob`p
p
u
d
zD†
-'(&+m"-::"†
R+$:>'(&†
!
t
= +
= − +
= −
x t
y
z t
M-#"<8/+O†
⇒
bnpUnpQtd
=
uuur
BH
bnpU
t
pUtd
+8
* `=
uur r
BI u
⇔
nqnU
t
nao`
⇔
o
x
−
⇒
uuur
BH
ob
t
q
p
qt
x
−
p
t
x
do
x
b_pUqtptdo
x
r
u
6(&8"
y ~
α
=
uur uur uur
u u n
obq`pap_ad
R!
q` a _a
= =
x-1 y+1 z -2
aeB"8"%-FG#*2"4#*f
(Ef
3"`#^%a2(E>")#*<8#*<8#*
(;%#"'\%&23"F]#*2J #"!123"4#*567$"Wf
(E2R8(;%f
!P2*+$
/;##"M2
G%*%N%
jˆ$ '(&=3‡k
I
::"†
"-\†
h*MT-'0$
>'.k
I
"--#"<8/+T$bid3‰Tv
⊥
†
M8]+bid"-k
)
-8
·
IMH
+$ "< hv 8 :
·
IMH
o
≥
HM MJ
IM IM
3<'P
^1+8
·
IMH
=3hvohT+1…
v3'8
·
IMH
obk
(
ik
)
l"-bid-$%&;+
k
(
'L
("<8"$%&bk
(
ik
)
l
V'8bidI
y y ~~
∆ ∆ ∆
r r r
u u u
-$"D
1#*
Gia?@AB8CD"89>5E5 F> GH1 Trang
25/33
