Bài tập toán cao cấp part 10 pps

9.2. Vi phˆan cu
’
a h`am nhiˆe
`
ubiˆe
´
n 143
27. w = f(ax, by,cz).
(D
S. d
n
w =

a
∂
∂α
dx + b
∂
∂β
dy + c
∂
∂γ
dz

n
f(α,β,γ),
α = ax, β = by, γ = cz)
Khai triˆe
’
n c´ac h`am d
˜a cho theo cˆong th´u

.
c Taylor d
ˆe
´
n c´ac sˆo
´
ha
.
ng
cˆa
´
p 2 (28-30) nˆe
´
u
28. f(x, y)=
1
x − y
(D
S. ∆w =
∆y − ∆x
(x − y)
2
+
∆x
2
− 2∆x∆y +∆y
2
(x −y)
3
+ R

2
)
29. f(x, y)=
√
x + y.
(DS. ∆w =
∆x +∆y
2
√
x + y
−
∆x
2
+2∆x∆y +∆y
2
8(x + y)
3/2
+ R
2
)
30. f(x, y)=e
x+y
.
D
S. ∆w = e
x+y
(∆x +∆y)+e
x+y
(∆x +∆y)
2

2
+ R
2
).
´
Ap du
.
ng vi phˆan d
ˆe
’
t´ınh gˆa
`
nd´ung (31-35)
31. i) a =(0,97)
2,02
(DS. ≈ 0, 94)
ii) b =

(4, 05)
2
+(2,93)
2
(DS. ≈ 4.998)
32. i) a =

(1.04)
2,99
+ln1, 02. (DS. 1,05)
Chı
’

dˆa
˜
n. X´et h`am
√
x
y
+lnz.
ii) b =
3

(1, 02)
2
+(0,05)
2
.(DS. 1,013)
Chı
’
dˆa
˜
n. X´et h`am
3

x
2
+ y
2
.
33. i) a = sin 29
◦
· tg46

◦
.(DS. ≈ 0, 502)
ii) b = sin 32
◦
· cos 59
◦
.(DS. ≈ 0, 273)
34. i) a = ln(0, 09
3
+0, 99
3
). (DS. ≈−0, 03)
144 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
Chı
’
dˆa
˜
n. X´et h`am f = ln(x
3
+ y
3
), M

0
(0, 1).
ii) b =

5e
0,02
+(2,03)
2
.(DS. ≈ 3, 037)
Chı
’
dˆa
˜
n. X´et h`am f =

5e
x
+ y
2
, M
0
(0, 2).
35. T´ınh vi phˆan cu
’
a h`am f(x, y)=

x
3
+ y
3

.
´
U
.
ng du
.
ng d
ˆe
’
t´ınh
xˆa
´
pxı
’

(1, 02)
3
+(1,97)
3
.(DS. ≈ 2, 95)
Trong c´ac b`ai to´an sau dˆay (36-38) h˜ay t´ınh vi phˆan cˆa
´
p1cu
’
a
h`am ˆa
’
n z(x, y) x´ac d
i
.

nh bo
.
’
i c´ac phu
.
o
.
ng tr`ınh tu
.
o
.
ng ´u
.
ng
36. z
3
+3x
2
z =2xy.(DS. dz =
(2y − 6xz)dx +2xdy
3(x
2
+ z
2
)
)
37. cos
2
x + cos
2

y + cos
2
z =1.
(D
S. dz = −
sin 2xdx + sin 2ydy
sin 2z
).
38. x + y + z = e
−(x+y+z)
.(DS. dz = −dx −dy)
39. Cho w l`a h`am cu
’
a x v`a y x´ac di
.
nh bo
.
’
iphu
.
o
.
ng tr`ınh
x
w
=ln
w
y
+1.
T´ınh vi phˆan dw, d

2
w.
(DS. dw =
w(ydx + wdy)
y(x + w)
,d
2
w = −
w
2
(ydx −xdy)
2
y
2
(x + w)
2
).
40. T´ınh dw v`a d
2
w nˆe
´
u h`am w(x, y)du
.
o
.
.
c x´ac di
.
nh bo
.

’
iphu
.
o
.
ng tr`ınh
w − x = arctg
y
w − x
.
(D
S. dw = dx +
(w − x)dy
(w − x)
2
+ y
2
+ y
,
d
2
w = −
2(y + 1)(w − x)[(w − x)
2
+ y
2
]
[(w − x)
2
+ y

2
+ y]
3
dy
2
).
9.3. Cu
.
.
c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n 145
9.3 Cu
.
.
c tri
.
cu
’
ah`am nhiˆe
`
ubiˆe
´
n

9.3.1 Cu
.
.
c tri
.
H`am f(x, y) c´o cu
.
.
cd
a
.
idi
.
aphu
.
o
.
ng (ho˘a
.
ccu
.
.
ctiˆe
’
ud
i
.
aphu
.
o

.
ng) b˘a
`
ng
f(x
0
,y
0
)ta
.
idiˆe
’
m M
0
(x
0
,y
0
) ∈ D nˆe
´
utˆo
`
nta
.
i δ-lˆan cˆa
.
ncu
’
adiˆe
’

m M
0
sao cho v´o
.
imo
.
idiˆe
’
m M = M
0
thuˆo
.
c lˆan cˆa
.
nˆa
´
y ta c´o
f(M) <f(M
0
) (tu
.
o
.
ng ´u
.
ng : f( M) >f(M
0
)).
Go
.

i chung cu
.
.
cda
.
i, cu
.
.
ctiˆe
’
ucu
’
a h`am sˆo
´
l`a cu
.
.
c tri
.
cu
’
a h`am sˆo
´
.
Diˆe
`
ukiˆe
.
ncˆa
`

ndˆe
’
tˆo
`
nta
.
icu
.
.
c tri
.
di
.
aphu
.
o
.
ng: Nˆe
´
uta
.
idiˆe
’
m M
0
h`am
f(x, y)c´ocu
.
.
c tri

.
d
i
.
aphu
.
o
.
ng th`ı ta
.
id
iˆe
’
md´oca
’
hai da
.
o h`am riˆeng cˆa
´
p
1(nˆe
´
uch´ung tˆo
`
nta
.
i) dˆe
`
ub˘a
`

ng 0 ho˘a
.
c ´ıt nhˆa
´
tmˆo
.
t trong hai da
.
o h`am
riˆeng khˆong tˆo
`
nta
.
i(d´o l`a nh˜u
.
ng diˆe
’
mt´o
.
iha
.
n ho˘a
.
c diˆe
’
md`u
.
ng cu
’
a

h`am f(x, y)). Khˆong pha
’
imo
.
idiˆe
’
md`u
.
ng dˆe
`
ul`adiˆe
’
mcu
.
.
c tri
.
.
D
iˆe
`
ukiˆe
.
ndu
’
: gia
’
su
.
’

f

xx
(M
0
)=,f

xy
(M
0
)=B, f

yy
(M
0
)=C.
Khi d´o:
i) Nˆe
´
u∆(M
0
)=





AB
BC






> 0v`aA>0th`ıta
.
id
iˆe
’
m M
0
h`am f c´o
cu
.
.
ctiˆe
’
ud
i
.
aphu
.
o
.
ng.
ii) Nˆe
´
u∆(M
0
)=






AB
BC





> 0v`aA<0th`ıta
.
id
iˆe
’
m M
0
h`am f c´o
cu
.
.
cda
.
idi
.
aphu
.
o

.
ng.
iii) Nˆe
´
u∆(M
0
)=





AB
BC





< 0th`ıM
0
l`a diˆe
’
m yˆen ngu
.
.
acu
’
a f,t´u
.

c
l`a ta
.
i M
0
h`am f khˆong c´o cu
.
.
c tri
.
.
iv) Nˆe
´
u∆(M
0
)=





AB
BC





=0th`ıM
0

l`a diˆe
’
m nghi vˆa
´
n (h`am f c´o
thˆe
’
c´o v`a c˜ung c´o thˆe
’
khˆong c´o cu
.
.
c tri
.
ta
.
id´o).
146 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
9.3.2 Cu
.
.
c tri

.
c´o d
iˆe
`
ukiˆe
.
n
Trong tru
.
`o
.
ng ho
.
.
pd
o
.
n gia
’
n nhˆa
´
t, cu
.
.
c tri
.
c´o d
iˆe
`
ukiˆe

.
ncu
’
a h`am f(x, y)
l`a cu
.
.
cda
.
i ho˘a
.
ccu
.
.
ctiˆe
’
ucu
’
a h`am d´oda
.
tdu
.
o
.
.
cv´o
.
idiˆe
`
ukiˆe

.
n c´ac biˆe
´
n
x v`a y tho
’
a m˜an phu
.
o
.
ng tr`ınh ϕ(x, y)=0(phu
.
o
.
ng tr`ınh r`ang buˆo
.
c).
D
ˆe
’
t`ım cu
.
.
c tri
.
c´o diˆe
`
ukiˆe
.
nv´o

.
idiˆe
`
ukiˆe
.
n r`ang buˆo
.
c ϕ(x, y) ta lˆa
.
p
h`am Lagrange (h`am bˆo
’
tro
.
.
)
F (x, y)=f(x, y)
λ
ϕ(x, y)
trong d´o λ l`a h˘a
`
ng sˆo
´
nhˆan chu
.
ad
u
.
o
.

.
c x´ac d
i
.
nh v`a di t`ım cu
.
.
c tri
.
thˆong
thu
.
`o
.
ng cu
’
a h`am bˆo
’
tro
.
.
n`ay. D
ˆa y l `a phu
.
o
.
ng ph´ap th`u
.
asˆo
´

bˆa
´
td
i
.
nh
Lagrange.
T`ım d
iˆe
`
ukiˆe
.
ncˆa
`
ndˆe
’
tˆo
`
nta
.
icu
.
.
c tri
.
c´o d
iˆe
`
ukiˆe
.

n chung quy l`a gia
’
i
hˆe
.
phu
.
o
.
ng tr`ınh













∂F
∂x
=
∂f
∂x
+ λ
∂ϕ

∂x
=0
∂F
∂y
=
∂f
∂y
+ λ
∂ϕ
∂y
=0
ϕ(x, y)=0
(9.15)
T`u
.
hˆe
.
n`ay ta c´o thˆe
’
x´ac di
.
nh x, y v`a λ.
Vˆa
´
nd
ˆe
`
tˆo
`
nta

.
iv`ad˘a
.
c t´ınh cu
’
acu
.
.
c tri
.
di
.
aphu
.
o
.
ng du
.
o
.
.
c minh di
.
nh
trˆen co
.
so
.
’
x´et dˆa

´
ucu
’
a vi phˆan cˆa
´
p hai cu
’
a h`am bˆo
’
tro
.
.
d
2
F =
∂
2
F
∂x
2
dx
2
+2
∂
2
F
∂x∂y
dxdy +
∂
2

F
∂y
2
dy
2
du
.
o
.
.
c t´ınh dˆo
´
iv´o
.
i c´ac gi´a tri
.
x, y, λ thu du
.
o
.
.
c khi gia
’
ihˆe
.
(9.15) v´o
.
idiˆe
`
u

kiˆe
.
nl`a
∂ϕ
∂x
dx +
∂ϕ
∂y
dy =0 (dx
2
+ dy
2
=0).
Cu
.
thˆe
’
l`a:
9.3. Cu
.
.
c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n 147

i) Nˆe
´
u d
2
F<0 h`am f(x, y) c´o cu
.
.
cd
a
.
ic´odiˆe
`
ukiˆe
.
n.
ii) Nˆe
´
u d
2
F>0 h`am f(x, y) c´o cu
.
.
ctiˆe
’
uc´od
iˆe
`
ukiˆe
.
n.

iii) Nˆe
´
u d
2
F = 0 th`ı cˆa
`
n pha
’
i kha
’
o s´at.
Nhˆa
.
nx´et
i) Viˆe
.
c t`ım cu
.
.
c tri
.
cu
’
a h`am ba biˆe
´
n ho˘a
.
c nhiˆe
`
uho

.
ndu
.
o
.
.
ctiˆe
´
n h`anh
tu
.
o
.
ng tu
.
.
nhu
.
o
.
’
1.
ii) Tu
.
o
.
ng tu
.
.
c´o thˆe

’
t`ım cu
.
.
c tri
.
c´o diˆe
`
ukiˆe
.
ncu
’
a h`am ba biˆe
´
n ho˘a
.
c
nhiˆe
`
uho
.
nv´o
.
imˆo
.
t ho˘a
.
c nhiˆe
`
uphu

.
o
.
ng tr`ınh r`ang buˆo
.
c(sˆo
´
phu
.
o
.
ng
tr`ınh r`ang buˆo
.
c pha
’
ib´eho
.
nsˆo
´
biˆe
´
n). Khi d´ocˆa
`
nlˆa
.
p h`am bˆo
’
tro
.

.
v´o
.
i
sˆo
´
th `u
.
asˆo
´
chu
.
ax´acd
i
.
nh b˘a
`
ng sˆo
´
phu
.
o
.
ng tr`ınh r`ang buˆo
.
c.
iii) Ngo`ai phu
.
o
.

ng ph´ap th`u
.
asˆo
´
bˆa
´
tdi
.
nh Lagrange, ngu
.
`o
.
i ta c`on
d`ung phu
.
o
.
ng ph´ap khu
.
’
biˆe
´
nsˆo
´
dˆe
’
t`ım cu
.
.
c tri

.
c´o diˆe
`
ukiˆe
.
n.
9.3.3 Gi´a tri
.
l´o
.
n nhˆa
´
t v`a b´e nhˆa
´
tcu
’
a h`am
H`am kha
’
vi trong miˆe
`
nd´ong bi
.
ch˘a
.
nda
.
t gi´a tri
.
l´o

.
n nhˆa
´
t (nho
’
nhˆa
´
t)
ho˘a
.
cta
.
idiˆe
’
md`u
.
ng ho˘a
.
cta
.
idiˆe
’
m biˆen cu
’
amiˆe
`
n.
C
´
AC V

´
IDU
.
V´ı du
.
1. T`ım cu
.
.
c tri
.
di
.
aphu
.
o
.
ng cu
’
a h`am
f(x, y)=x
4
+ y
4
− 2x
2
+4xy − 2y
2
.
Gia
’

i. i) Miˆe
`
n x´ac d
i
.
nh cu
’
a h`am l`a to`an m˘a
.
t ph˘a
’
ng R
2
.
ii) T´ınh c´ac da
.
o h`am riˆeng f

x
v`a f

y
v`a t`ım c´ac diˆe
’
mt´o
.
iha
.
n. Ta
c´o

f

x
=4x
3
− 4x +4y, f

y
=4y
3
+4x −4y.
Do d
´o
4x
3
− 4x +4y =0
4y
3
+4x −4y =0
148 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
v`a t `u
.

d
´o

x
1
=0
y
1
=0

x
2
= −
√
2
y
2
=
√
2

x
3
=
√
2
y
3
= −
√

2.
Nhu
.
vˆa
.
y ta c´o ba diˆe
’
mt´o
.
iha
.
n. V`ı f

x
, f

y
tˆo
`
nta
.
iv´o
.
imo
.
idiˆe
’
m
M(x, y) ∈ R
2

nˆen h`am khˆong c`on diˆe
’
mt´o
.
iha
.
n n`ao kh´ac.
iii) Ta t´ınh c´ac d
a
.
o h`am riˆeng cˆa
´
p hai v`a gi´a tri
.
cu
’
ach´ung ta
.
i c´ac
diˆe
’
mt´o
.
iha
.
n.
f

xx
(x, y)=12x

2
=4,f

xy
=4,f

yy
=12y
2
− 4.
Ta
.
id
iˆe
’
m O(0, 0): A = −4, B =4,C = −4
Ta
.
idiˆe
’
m M
1
(−
√
2, +
√
2): A = 20, B =4,C =20
Ta
.
id

iˆe
’
m M
2
(+
√
2, −
√
2): A = 20, B =4,C = 20.
iv) Ta
.
idiˆe
’
m O(0, 0)ta c´o





AB
BC





=






−44
4 −4





=16−16 = 0.
Dˆa
´
uhiˆe
.
ud
u
’
khˆong cho ta cˆau tra
’
l`o
.
i. Ta nhˆa
.
n x´et r˘a
`
ng trong lˆan
cˆa
.
nbˆa
´

tk`ycu
’
adiˆe
’
mOtˆo
`
nta
.
inh˜u
.
ng diˆe
’
mm`af(x, y) > 0v`anh˜u
.
ng
d
iˆe
’
mm`af( x, y) < 0. Ch˘a
’
ng ha
.
ndo
.
c theo trung c Ox (y = 0) ta c´o
f(x, y)


y=0
= f(x, 0) = x

4
− 2x
2
= −x
2
(2 − x
2
) < 0
ta
.
inh˜u
.
ng diˆe
’
mdu
’
gˆa
`
n(0, 0), v`a do
.
c theo du
.
`o
.
ng th˘a
’
ng y = x
f(x, y)



y=x
= f(x, x)=2x
4
> 0
Nhu
.
vˆa
.
y, ta
.
inh˜u
.
ng d
iˆe
’
m kh´ac nhau cu
’
amˆo
.
t lˆan cˆa
.
n n`ao d´ocu
’
a
diˆe
’
m O(0, 0) sˆo
´
gia to`an phˆa
`

n∆f(x, .y) khˆong c´o c`ung mˆo
.
tdˆa
´
u v`a do
d´o t a
.
i O(0, 0) h`am khˆong c´o cu
.
.
c tri
.
di
.
aphu
.
o
.
ng.
Ta
.
idiˆe
’
m M
1
(−
√
2,
√
2) ta c´o






AB
BC





=





20 4
420





= 400 − 16 > 0
9.3. Cu
.
.
c tri

.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n 149
v`a A>0nˆen ta
.
i M
1
(−
√
2,
√
2) h`am c´o cu
.
.
ctiˆe
’
ud
i
.
aphu
.
o
.
ng v`a
f

min
= −8.
Ta
.
idiˆe
’
m M
2
(
√
2, −
√
2) ta c´o AC − B
2
> 0v`aA>0 nˆen ta
.
id´o
h`am c´o cu
.
.
ctiˆe
’
udi
.
aphu
.
o
.
ng v`a f
min

= −8.
V´ı du
.
2. Kha
’
o s´at v`a t`ım cu
.
.
c tri
.
cu
’
a h`am
f(x, y)=x
2
+ xy + y
2
− 2x − 3y.
Gia
’
i. i) Hiˆe
’
n nhiˆen D
f
≡ R.
ii) T`ım diˆe
’
md`u
.
ng. Ta c´o

f

x
=2x + y − 2
f

y
= x +2y − 3
⇒
2x + y − 2=0,
x +2y − 3=0.
Hˆe
.
thu d
u
.
o
.
.
c c´o nghiˆe
.
ml`ax
0
=
1
3
, y
0
=
4

3
.Dod´o

1
3
,
4
3

l`a diˆe
’
m
d`u
.
ng v`a ngo`ai diˆe
’
md`u
.
ng d´o h`am f khˆong c´o diˆe
’
md`u
.
ng n`ao kh´ac v`ı
f

x
v`a f

y
tˆo

`
ntˆa
.
i ∀(x, y).
iii) Kha
’
o s´at cu
.
.
c tri
.
.Tac´oA = f

x
2
=2,Bf

xy
=1,C = f

y
2
=2.
Do d
´o
∆(M
0
)=






21
12





=3> 0v`aA =2> 0
nˆen h`am f c´o cu
.
.
ctiˆe
’
uta
.
id
iˆe
’
m M
0
(
1
3
,
4
3


. 
V´ı du
.
3. T`ım cu
.
.
c tri
.
cu
’
a h`am f(x, y)=6− 4x −3y v´o
.
idiˆe
`
ukiˆe
.
nl`a
x v`a y liˆen hˆe
.
v´o
.
i nhau bo
.
’
iphu
.
o
.
ng tr`ınh x
2

+ y
2
=1.
Gia
’
i. Ta lˆa
.
p h`am Lagrange
F (x, y)=6−4x − 3y + λ(x
2
+ y
2
− 1).
Ta c´o
∂F
∂x
= −4+2λx,
∂F
∂y
= −3+2λy
150 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
v`a ta gia

’
ihˆe
.
phu
.
o
.
ng tr`ınh
−4+2λx =0
−3+2λx =0
x
2
+ y
2
=1
Gia
’
i ra ta c´o
λ
1
=
5
2
,x
1
=
4
5
,y
1

=
3
5
λ
2
= −
5
2
,x
2
= −
4
5
,y
2
= −
3
5
V`ı
∂
2
F
∂x
2
=2λ,
∂
2
F
∂x∂y
=0,

∂
2
F
∂y
2
=2λ
nˆen
d
2
F =2λ(dx
2
+ dy
2
).
Nˆe
´
u λ =
5
2
, x =
4
5
, y =
3
5
th`ı d
2
F>0nˆenta
.
idiˆe

’
m

4
5
,
3
5

h`am
c´o cu
.
.
ctiˆe
’
uc´odiˆe
`
ukiˆe
.
n.
Nˆe
´
u λ = −
5
2
, x = −
4
5
, y = −
3

5
th`ı d
2
F<0v`adod´o h`am c´o cu
.
.
c
d
a
.
ic´odiˆe
`
ukiˆe
.
nta
.
idiˆe
’
m

−
4
5
, −
3
5

.
Nhu
.

vˆa
.
y
f
max
=6+
16
5
+
9
5
=11,
f
min
=6−
16
5
−
9
5
=1. 
V´ı d u
.
4. T`ım cu
.
.
c tri
.
c´o diˆe
`

ukiˆe
.
ncu
’
a h`am
1) f(x, y)=x
2
+ y
2
+ xy − 5x − 4y +10,x+ y =4.
2) u = f(x, y, z)=x + y + z
2

z − x =1,
y − xz =1.
9.3. Cu
.
.
c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n 151
Gia
’
i. 1) T`u

.
phu
.
o
.
ng tr`ınh r`ang buˆo
.
c x + y =4tac´oy =4−x v`a
f(x, y)=x
2
+(4−x)
2
+ x(4 − x) −5x −4(4 − x)+10
= x
2
− 5x +10,
ta thu d
u
.
o
.
.
c h`am mˆo
.
tbiˆe
´
nsˆo
´
g(x)=x
2

− 5x +10
v`a cu
.
.
c tri
.
di
.
aphu
.
o
.
ng cu
’
a g(x)c˜ung ch´ınh l`a cu
.
.
c tri
.
c´o diˆe
`
ukiˆe
.
ncu
’
a
h`am f(x, y).
´
Ap du
.

ng phu
.
o
.
ng ph´ap kha
’
o s´at h`am sˆo
´
mˆo
.
tbiˆe
´
nsˆo
´
d
ˆo
´
i
v´o
.
i g(x) ta t`ım du
.
o
.
.
c g(x) c´o cu
.
.
ctiˆe
’

udi
.
aphu
.
o
.
ng
g
min
= g

5
2

=
15
4
·
Nhu
.
ng khi d
´o h `a m f(x, y)d˜a cho c´o
cu
.
.
ctiˆe
’
uc´odiˆe
`
ukiˆe

.
nta
.
idiˆe
’
m

5
2
,
3
2

(y =4− x ⇒ y =4−
5
2
=
3
2
)v`a
f
min
= f

5
2
,
3
2


=
15
4
·
2) T`u
.
c´ac phu
.
o
.
ng tr`ınh r`ang buˆo
.
c ta c´o
z =1+x
y = x
2
+ x +1
v`a thˆe
´
v`ao h`am d˜a cho ta du
.
o
.
.
c h`am mˆo
.
tbiˆe
´
nsˆo
´

u = f(x, y(x),z(x)) = g(x)=2x
2
+4x +2.
Dˆe
˜
d`ang thˆa
´
yr˘a
`
ng h`am g(x)c´ocu
.
.
ctiˆe
’
uta
.
i x = −1 (khi d´o y =1,
z = 0) v`a do d
´o h`am f(x, y, z) c´o cu
.
.
ctiˆe
’
uc´od
iˆe
`
ukiˆe
.
nta
.

idiˆe
’
m
(−1, 1, 0) v`a
f
min
= f(−1, 1, 0)=0. 
152 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
V´ı du
.
5. B˘a
`
ng phu
.
o
.
ng ph´ap th`u
.
asˆo
´
bˆa
´

td
i
.
nh Lagrange t`ım cu
.
.
c tri
.
c´o d
iˆe
`
ukiˆe
.
ncu
’
a h`am
u = x + y + z
2
v´o
.
id
iˆe
`
ukiˆe
.
n

z − x =1
y − xz =1
(9.16)

(xem v´ı du
.
4, ii)).
Gia
’
i. Ta lˆa
.
p h`am Lagrange
F (x, y, z)=x + y + z
2
+ λ
1
(z − x − 1) + λ
2
(y − zx − 1)
v`a x´et hˆe
.
phu
.
o
.
ng tr`ınh
























∂F
∂x
=1−λ
1
− λ
2
z =0
∂F
∂y
=1+λ
2
=0
∂F
∂z
=2z + λ

1
−λ
2
x =0
ϕ
1
= z − x − 1=0
ϕ
2
= y − xz − 1=0.
Hˆe
.
n`ay c´o nghiˆe
.
m duy nhˆa
´
t x = −1, y =1,z =0,λ
1
=1v`a
λ
2
= −1 ngh˜ıa l`a M
0
(−1, 1, 0) l`a diˆe
’
m duy nhˆa
´
t c´o thˆe
’
c´o cu

.
.
c tri
.
cu
’
a
h`am v´o
.
i c´ac diˆe
`
ukiˆe
.
n r`ang buˆo
.
c ϕ
1
v`a ϕ
2
.
T`u
.
c´ac hˆe
.
th ´u
.
c
z − x =1
y − xz =1
ta thˆa

´
yr˘a
`
ng (9.16) x´ac d
i
.
nh c˘a
.
p h`am ˆa
’
n y(x)v`az(x) (trong tru
.
`o
.
ng
ho
.
.
p n`ay y(x)v`az(x)dˆe
˜
d`ang r´ut ra t`u
.
(9.16)). Gia
’
su
.
’
thˆe
´
nghiˆe

.
m
9.3. Cu
.
.
c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n 153
y(x)v`az(x) v`ao hˆe
.
(9.16) v`a b˘a
`
ng c´ach lˆa
´
y vi phˆan c´ac dˆo
`
ng nhˆa
´
t
th ´u
.
cthud
u
.

o
.
.
c ta c´o

dz − dx =0
dy − xdz −zdx =0
⇒

dz = dx
dy =(x + z) dx.
(9.17)
Bˆay gi`o
.
t´ınh vi phˆan cˆa
´
p hai cu
’
a h`am Lagrange
d
2
F =2(dz)
2
− 2λ
2
dxdz. (9.18)
Thay gi´a tri
.
λ
2

= −1 v`a (9.17) v`ao (9.18) ta thu du
.
o
.
.
cda
.
ng to`an
phu
.
o
.
ng x´ac di
.
nh du
.
o
.
ng l`a
d
2
F =4dx
2
.
T`u
.
d´o suy ra h`am d˜a cho c´o cu
.
.
ctiˆe

’
uc´odiˆe
`
ukiˆe
.
nta
.
idiˆe
’
m
M
0
(−1, 1, 0) v`a f
min
=0. 
V´ı du
.
6. T`ım gi´a tri
.
l´o
.
n nhˆa
´
t v`a nho
’
nhˆa
´
tcu
’
a h`am

f(x, y)=x
2
+ y
2
− xy + x + y
trong miˆe
`
n
D = {x  0,y  0,x+ y  −3}.
Gia
’
i. Miˆe
`
n D d
˜a cho l`a tam gi´ac OAB v´o
.
idı
’
nh ta
.
i A(−3, 0),
B(0, −3) v`a O(0, 0).
i) T`ım c´ac d
iˆe
’
md`u
.
ng:
f


x
=2x −y +1=0
f

y
=2y − x +1=0
T`u
.
d
´o x = −1, y = −1. Vˆa
.
ydiˆe
’
md`u
.
ng l`a M(−1, −1).
Ta
.
id
iˆe
’
m M ta c´o:
f(M)=f(−1, −1) = −1.
154 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe

´
n
ii) Ta c´o
A = f

xx
(−1, −1)=2
B = f

xy
(−1, −1) = −1
C = f

yy
(−1, −1) = 2.
Vˆa
.
y AC −B
2
=4−1=3> 0, nˆen h`am c´o biˆe
.
tth´u
.
c AC −B
2
> 0
v`a A =2> 0. Do d
´o t a
.
idiˆe

’
m M n´o c´o cu
.
.
ctiˆe
’
ud
i
.
aphu
.
o
.
ng v`a
f
min
= −1.
iii) Kha
’
o s´at h`am trˆen biˆen cu
’
amiˆe
`
n D.
+) Khi x =0tac´of = y
2
+ y.Dˆo
´
iv´o
.

i h`am mˆo
.
tbiˆe
´
n f = y
2
+ y,
−3  y  0 ta c´o
(f
ln
)


x=0
=6ta
.
idiˆe
’
m(0, −3)
(f
nn
)


x=0
=
−1
4
ta
.

idiˆe
’
m

0, −
1
2

.
+) Khi y = 0 ta c´o h`am mˆo
.
tbiˆe
´
n f = x
2
+ x, −3  x  0v`a
tu
.
o
.
ng tu
.
.
:
(f
ln
)


y=0

=6ta
.
idiˆe
’
m(0, −3)
(f
nn
)


y=0
=
−1
4
ta
.
idiˆe
’
m

−
1
2
, 0

.
+) Khi x + y = −3 ⇒ y = −3 − x ta c´o f(x)=3x
2
+9x +6v`a
(f

nn
)


x+y=−3
=
−3
4
ta
.
idiˆe
’
m

−
3
2
, −
3
2

(f
ln
)


x+y=−3
=6ta
.
idiˆe

’
m(0, −3) v`a (−3, 0).
iv) So s´anh c´ac gi´a tri
.
thu du
.
o
.
.
cdˆo
´
iv´o
.
i f ta kˆe
´
t luˆa
.
n f
ln
=6ta
.
i
(0, −3) v`a (−3, 0) v`a gi´a tri
.
f
nn
= −1ta
.
idiˆe
’

md`u
.
ng (−1, −1).
B
`
AI T
ˆ
A
.
P
9.3. Cu
.
.
c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe
´
n 155
H˜ay t`ım cu
.
.
c tri
.
cu
’
a c´ac h`am sau d

ˆay
1. f =1+6x − x
2
− xy − y
2
.(DS. f
max
= 13 ta
.
idiˆe
’
m(4, −2))
2. f =(x −1)
2
+2y
2
.(DS. f
min
=0ta
.
idiˆe
’
m(1, 0))
3. f = x
2
+ xy + y
2
− 2x − y.(DS. f
min
= −1ta

.
idiˆe
’
m(1, 0))
4. f = x
3
y
2
(6−x−y)(x>0,y >0). (DS. f
max
= 108 ta
.
idiˆe
’
m(3, 2))
5. f =2x
4
+ y
4
− x
2
− 2y
2
.
(D
S. f
max
=0ta
.
idiˆe

’
m(0, 0),
f
min
= −
9
8
ta
.
ic´acdiˆe
’
m M
1

1
2
, −1

v`a M
2

−1
2
, 1

f
min
= −
9
8

ta
.
ic´acdiˆe
’
m M
3

1
2
, −1

v`a M
4

−1
2
, −1

)
6. f =(5x +7y − 25)e
−(x
2
+xy+y
2
)
.
(DS. f
max
=3
−13

ta
.
idiˆe
’
m M
1
(1, 3),
f
min
= −26e
−1/52
ta
.
idiˆe
’
m M
2

−1
26
,
−3
26

)
7. f = xy +
50
x
+
20

y
, x>0, y>0. (DS. f
min
= 30 ta
.
idiˆe
’
m(5, 2))
8. f = x
2
+ xy + y
2
− 6x − 9y.(DS. f
min
= −21 ta
.
idiˆe
’
m(1, 4))
9. f = x
√
y − x
2
− y +6x + 3. (DS. f
max
= 15 ta
.
idiˆe
’
m(4, 4))

10. f =(x
2
+ y)
√
e
y
.(DS. f
min
= −
2
e
ta
.
i(0, −2))
11. f =2+(x − 1)
4
(y +1)
6
.(DS. f
min
=2ta
.
idiˆe
’
m(1, −1))
Chı
’
dˆa
˜
n. Ta

.
idiˆe
’
m M
0
(1, −1) ta c´o ∆(M
0
)=0.Cˆa
`
n kha
’
o s´at dˆa
´
u
cu
’
a f(M) −f(M
0
)=f(1 + ∆x, −1+∆y) −f(1, −1).
12. f =1− (x − 2)
4/5
− y
4/5
.(DS. f
max
=1ta
.
idiˆe
’
m(2, 0))

Chı
’
dˆa
˜
n. Ta
.
idiˆe
’
m(2, 0) h`am khˆong kha
’
vi. Kha
’
o s´at dˆa
´
ucu
’
a
f(M) − f(M
0
), M
0
=(2,0).
156 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´

n
T`ım cu
.
.
c tri
.
c´o d
iˆe
`
ukiˆe
.
ncu
’
a c´ac h`am sau dˆay
13. f = xy v´o
.
id
iˆe
`
ukiˆe
.
n x + y =1.
(DS. f
max
=
1
4
ta
.
idiˆe

’
m

1
2
,
1
2

)
14. f = x +2y v´o
.
id
iˆe
`
ukiˆe
.
n x
2
+ y
2
=5.
(DS. f
max
=5ta
.
idiˆe
’
m(1, 2))
15. f = x

2
+ y
2
v´o
.
idiˆe
`
ukiˆe
.
n
x
2
+
y
3
=1.
(DS. f
min
=
36
13
ta
.
idiˆe
’
m

18
13
,

12
13

)
16. f = x − 2y +2z v´o
.
id
iˆe
`
ukiˆe
.
n x
2
+ y
2
+ z
2
=9.
(DS. f
min
= −9ta
.
idiˆe
’
m(−1, 2, −2); f
max
=9ta
.
i(1, −2, 2).)
17. f = xy v´o

.
idiˆe
`
ukiˆe
.
n2x +3y =5.
(DS. f
max
=
25
24
ta
.
idiˆe
’
m

5
4
,
5
6

)
18. 1) f = x
2
+ y
2
v´o
.

id
iˆe
`
ukiˆe
.
n r`ang buˆo
.
c
x
4
+
y
3
=1.
(D
S. f
min
=
144
25
ta
.
i

36
25
,
48
25


)
2) f = e
xy
v´o
.
idiˆe
`
ukiˆe
.
n x + y =1.
(DS. f
max
= e
1/4
ta
.
idiˆe
’
m

1
2
,
1
2

)
Chı
’
dˆa

˜
n. C´o thˆe
’
su
.
’
du
.
ng phu
.
o
.
ng ph´ap khu
.
’
biˆe
´
n.
19. f = x
2
+ y
2
+2z
2
v´o
.
idiˆe
`
ukiˆe
.

n x − y + z =1.
(DS. f
min
=0, 4ta
.
idiˆe
’
m(0, 4; −0, 4; 0, 2))
20. f = x
3
+ y
2
− z
3
+5v´o
.
id
iˆe
`
ukiˆe
.
n x + y −z =1.
(DS. f
min
=5ta
.
idiˆe
’
m(0,0, 0) v`a f
max

=7
10
27
ta
.
idiˆe
’
m

−
4
3
,
8
3
,
4
3

)
9.3. Cu
.
.
c tri
.
cu
’
a h`am nhiˆe
`
ubiˆe

´
n 157
21. f = xyz v´o
.
ic´acd
iˆe
`
ukiˆe
.
n x + y + z =5,xy + yz + zx =8.
(D
S. f
max
=4
4
27
ta
.
i

4
3
,
4
3
,
7
3

;


4
3
,
7
3
,
4
3

;

7
3
,
4
3
,
4
3

f
min
=4ta
.
i(2, 2, 1); (2, 1, 2); (1, 2, 2))
T`ım gi´a tri
.
l´o
.

n nhˆa
´
t v`a nho
’
nhˆa
´
tcu
’
a c´ac h`am sˆo
´
sau.
22. f = x
2
y(2 − x − y), D l`a tam gi´ac du
.
o
.
.
c gi´o
.
iha
.
nbo
.
’
ic´acd
oa
.
n
th˘a

’
ng x =0,y =0,x + y =6.
(D
S. f
ln
=
1
4
ta
.
idiˆe
’
m(1, 2); f
nn
= −128 ta
.
idiˆe
’
m(4, 2)).
23. f = x + y, D = {x
2
+ y
2
 1}.
(DS. f
ln
=
√
2ta
.

idiˆe
’
m biˆen

√
2
2
,
√
2
2

;
f
nn
= −
√
2ta
.
idiˆe
’
m biˆen

−
√
2
2
, −
√
2

2

).
24. T`u
.
mo
.
i tam gi´ac c´o chu vi b˘a
`
ng 2p, h˜ay t`ım tam gi´ac c´o diˆe
.
nt´ıch
l´o
.
n nhˆa
´
t.
Chı
’
dˆa
˜
n. D˘a
.
t a = x, b = y ⇒ c =2p − x − y v`a ´ap du
.
ng cˆong
th ´u
.
c Heron
S =


p(p − x)( p − y)(x + y − p)
(D
S. Tam gi´ac dˆe
`
u).
25. X´ac di
.
nh gi´a tri
.
l´o
.
n nhˆa
´
t v`a nho
’
nhˆa
´
tcu
’
a h`am
f = x
2
− y
2
,D= {x
2
+ y
2
 1}

(D
S. f
ln
=1ta
.
i(1, 0) v`a (−1, 0);
f
nn
= −1ta
.
i(0, 1) v`a (0, −1)).
26. X´ac di
.
nh gi´a tri
.
l´o
.
n nhˆa
´
t v`a nho
’
nhˆa
´
tcu
’
a h`am
f = x
3
− y
3

−3xy, D = {0  x  2, −1  y  2}.
158 Chu
.
o
.
ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe
`
ubiˆe
´
n
(DS. f
ln
= 13 ta
.
idiˆe
’
m(2, −1);
f
nn
= −1ta
.
idiˆe
’
m(1, 1) v`a (0, −1)).