BÀI GIẢNG GIẢI TÍCH HÀM NHIỀU BIẾN SỐ

TẬP ĐOÀN BƯU CHÍNH VIỄN THÔNG
HỌC VIỆN CÔNG NGHỆ BƯU CHÍNH VIỄN THÔNG










BÀI GIẢNG
GIẢI TÍCH HÀM NHIỀU BIẾN SỐ








PGS. TS. Phạm Ngọc Anh

HÀ NỘI-2013
PTIT
R
n
PTIT
PTIT
PTIT
PTIT
PTIT
R
n
x = (x
1
, x
2
, , x
n
) ∈ R
n
, y = (y
1
, y
2
, , y
n
) ∈ R

n
.
n R
n
x ± y = (x
1
± y
1
, x
2
± y
2
, , x
n
± y
n
).
λx = (λx
1
, λx
2
, , λx
n
), ∀λ ∈ R.
x, y = x
1
y
1
+ x
2

y
2
+ + x
n
y
n
.
x y x
1
y
1
+ x
2
y
2
+ + x
n
y
n
= 0.
x = 0 y = 0
cos(x, y) =
x
1
y
1
+ x
2
y
2

+ + x
n
y
n

x
2
1
+ x
2
2
+ + x
2
n

y
2
1
+ y
2
2
+ + y
2
n
.
x = (x
1
, x
2
, , x

n
) ∈ R
n
. x
x
x =

x
2
1
+ x
2
2
+ + x
2
n
.
x ≥ 0 ∀x ∈ R
n
x = 0 x = 0
λx = |λ|x ∀x ∈ R
n
, λ ∈ R
x + y ≤ x + y ∀x, y ∈ R
n
x ∈ R
n
y ∈ R
n
d(x, y) = x −y.

x ∈ R
n
, ǫ > 0
PTIT
B(x, ǫ) = {y ∈ R
n
: y −x < ǫ} x ǫ
¯
B(x, ǫ) = {y ∈ R
n
: y −x ≤ ǫ} x ǫ
x ∈ M ⊆ R
n
B(x, ǫ) B(x, ǫ) ⊆ M
M M intM
M ⊆ R
n
intM = M
M ⊆ R
n
x M ǫ > 0 B(x, ǫ)
M M M
∂M
M ⊆ R
n
∂M ⊆ M
M ⊆ R
n
α > 0 x ≤ α ∀x ∈ M
M ⊆ R

n
M
∅ = D ⊆ R
n
f : D → R
x = (x
1
, x
2
, , x
n
) ∈ D −→ y = f(x
1
, x
2
, , x
n
) ∈ R
D f
x
1
, x
2
, , x
n
f
R > 0
f(x) =

R

2
− x
2
1
− x
2
2
− −x
2
n
.
D
D ={x ∈ R
n
: R
2
− x
2
1
− x
2
2
− −x
2
n
≥ 0}
={x ∈ R
n
: x −0
2

≤ R
2
}
=
¯
B(0, R).
R
3
(S) = {(x, y, z) ∈ R
3
: (x −a)
2
+ (y −b)
2
+ (z −c)
2
= R
2
}.
PTIT
c
b
a
I
z
y
x
O
I(a, b, c) R (S)
(E) :

x
2
a
2
+
y
2
b
2
+
z
2
c
2
= 1 (a, b, c > 0).
(Oxy) z = 0 :
x
2
a
2
+
y
2
b
2
= 1.
(Oyz) x = 0 :
y
2
b

2
+
z
2
c
2
= 1.
(Oxz) y = 0 :
x
2
a
2
+
z
2
c
2
= 1.
(H
1
) :
x
2
a
2
+
y
2
b
2

−
z
2
c
2
= 1 (a, b, c > 0).
(Oxy) z = 0 :
x
2
a
2
+
y
2
b
2
= 1.
PTIT
c
b
O
x
y
z
a
(Oyz) x = 0 :
y
2
b
2

−
z
2
c
2
= 1.
(Oxz) y = 0 :
x
2
a
2
−
z
2
c
2
= 1.
(H
2
) :
x
2
a
2
+
y
2
b
2
−

z
2
c
2
= −1 (a, b, c > 0).
z
2
c
2
− 1 ≥ 0 ⇔ z ∈ (−∞, −c] ∪ [c, +∞).
(Oyz) x = 0 :
y
2
b
2
−
z
2
c
2
= −1.
(Oxz) y = 0 :
x
2
a
2
−
z
2
c

2
= −1.
(P ) z = h > c :
x
2
a
2
+
y
2
b
2
=
h
2
c
2
− 1.
PTIT
b
a
O
z
y
x
O
x
y
z
c

−c
(P E) :
x
2
p
+
y
2
q
= 2z (p, q > 0),
PTIT
z ≥ 0
(Oyz) x = 0 : y
2
= 2qz.
(Oxz) y = 0 : x
2
= 2pz.
(P ) z = h > 0 :
x
2
p
+
y
2
q
= 2h.
x
y
z

O
(T
z
) : f (x, y) = 0 Oz,
(T
y
) : g(x, z) = 0 Oy,
(T
x
) : f (y, z) = 0 Ox,
f, g, h : D ⊆ R
2
→ R
(N) :
x
2
a
2
+
y
2
b
2
−
z
2
c
2
= 0 (a, b, c > 0).
(Oyz) x = 0 : y = ±

b
c
z.
PTIT
O
z
y
x
Oz
(Oxz) y = 0 : x = ±
a
c
z.
(P ) z = h > 0 :
x
2
a
2
+
y
2
b
2
=
h
2
c
2
.
R

n
{M
n
} ⊂ R
2
M
0
∈ R
2
M
n
→ M
0
n → ∞ lim
n→∞
M
n
= M
0
ǫ > 0 n(ǫ)
M
n
∈ B(M
0
, ǫ) ∀n ≥ n(ǫ).
lim
n→∞
x
n
= x

0
lim
n→∞
y
n
= y
0
M
n
(x
n
, y
n
) → M
0
(x
0
, y
0
)
n → ∞
z = f (x, y) M
0
∈ R
2
M
0
m f(x, y) (x, y) M
0
(x

0
, y
0
)
lim
M→M
0
f(M) = m {M
n
} ⊂ R
2
lim
n→∞
M
n
= M
0
lim
n→∞
f(M
n
) = m.
lim
M→M
0
f(M) = m
∀ǫ > 0, ∃δ > 0, ∀M ∈ B( M
0
, δ) ⇒ |f (M) − m| < ǫ.
PTIT

x
y
z
O
I
1
= lim
(x,y)→(0,0)
x
2
y
2x
2
+ y
2
.
f(x, y) =
x
2
y
2x
2
+y
2
D = R
2
\{(0, 0)}
x
2
2x

2
+ y
2
≤
x
2
2x
2
=
1
2
∀(x, y) ∈ D,
|f(x, y)| ≤
1
2
|y| (x, y) ∈ D
0 ≤ lim
(x,y)→(0,0)




x
2
y
2x
2
+ y
2





≤ lim
(x,y)→(0,0)
1
2
|y| = 0.
I
1
= 0
I
2
= lim
(x,y)→(0,0)
xy
2x
2
+ y
2
.
f(x, y) =
xy
2x
2
+y
2
D = R
2
\{(0, 0)}

PTIT
(x, y) ∈ d : y = x (x, y) → (0, 0) x → 0
I
2
= lim
(x,y)→(0,0)
xy
2x
2
+ y
2
= lim
x→0
x
2
2x
2
+ x
2
=
1
3
.
(x, y) ∈ d : y = 3x (x, y) → (0, 0) x → 0
I
2
= lim
(x,y)→(0,0)
xy
2x

2
+ y
2
= lim
x→0
3x
2
2x
2
+ 9x
2
=
3
11
.
I
2
z = f (x, y) D M
0
∈ D
f M
0
lim
M→M
0
f(M) = f(M
0
).
f D f M ∈ D
f D ǫ > 0 δ > 0

∀(x, y), ( x
′
, y
′
) ∈ D : (x, y) − (x
′
, y
′
) < δ ⇒ |f(x, y) − f(x
′
, y
′
)| < ǫ.
f : D ⊆ R
2
→ R D f
D
f D D f D
f D f D
z = f (x) D ⊆ R
n
¯x = (¯x
1
, ¯x
2
, , ¯x
n
) ∈ D
x
1

−→ f (x
1
, ¯x
2
, , ¯x
n
) x
1
f x
1
¯x
f
′
x
1
(¯x)
∂f
∂x
1
(¯x).
f x
i
i = 1, 2, , n
¯x
f
′
x
i
(¯x)
∂f

∂x
i
(¯x).
PTIT
f(x, y) = x
2
tan(x
3
+ 2y).
∂f
∂x
= 2x tan(x
3
+ 2y) + x
2
.
1
cos
2
(x
3
+ 2y)
.3x
2
= 2x tan(x
3
+ 2y) +
3x
4
cos

2
(x
3
+ 2y)
.
∂f
∂y
= x
2
.
1
cos
2
(x
3
+ 2y)
.2 =
2x
2
cos
2
(x
3
+ 2y)
.
f : D ⊆ R
n
→ R ¯x = (¯x
1
, ¯x

2
, , ¯x
n
) ∈ D
x = (x
1
, x
2
, , x
n
) ∈ D
∆
x
i
= x
i
− ¯x
i
.
∆
f
= f(¯x
1
+ ∆
x
1
, ¯x
2
+ ∆
x

2
, , ¯x
n
+ ∆
x
n
) − f(¯x
1
, ¯x
2
, , ¯x
n
)
¯x
∆
f
= A
1
∆
x
1
+ A
2
∆
x
2
+ + A
n
∆
x

n
+ α
1
∆
x
1
+ α
2
∆
x
2
+ + α
n
∆
x
n
,
A
i
(i = 1, 2, , n) ¯x ∆
x
= (∆
x
1
, ∆
x
2
, , ∆
x
n

)
lim
∆
x
→0
α
k
= 0 ∀k = 1, 2, , n,
f ¯x
df = A
1
∆
x
1
+ A
2
∆
x
2
+ + A
n
∆
x
n
f ¯x
f D f ¯x ∈ D
f : D ⊆ R
n
→ R
¯x ∈ D f ¯x

df =
∂f
∂x
1
(¯x)∆
x
1
+
∂f
∂x
2
(¯x)∆
x
2
+ +
∂f
∂x
n
(¯x)∆
x
n
.
PTIT
∆
f
=f(¯x
1
+ ∆
x
1

, ¯x
2
+ ∆
x
2
, , ¯x
n
+ ∆
x
n
) −f(¯x
1
, ¯x
2
, , ¯x
n
)
=f(¯x
1
+ ∆
x
1
, ¯x
2
+ ∆
x
2
, , ¯x
n
+ ∆

x
n
) −f(¯x
1
, ¯x
2
+ ∆
x
2
, , ¯x
n
+ ∆
x
n
)
+ ···
+ f(¯x
1
, ¯x
2
, ¯x
n−1
, ¯x
n
+ ∆
x
n
) −f(¯x
1
, ¯x

2
, , ¯x
n
).
θ
1
, θ
2
, , θ
n
∈ (0, 1)
f(¯x
1
, , ¯x
i−1
, ¯x
i
+ ∆
x
i
, , ¯x
n
+ ∆
x
n
) − f(¯x
1
, , ¯x
i
, ¯x

i+1
+ ∆
x
i+1
, , ¯x
n
+ ∆
x
n
)
=
∂f
∂x
i
(¯x
1
, , ¯x
i−1
, ¯x
i
+ θ
i
∆
x
i
, , ¯x
n
+ ∆
x
n

)∆
x
i
.
¯x
∂f
∂x
i
(¯x
1
, , ¯x
i−1
, ¯x
i
+ θ
i
∆
x
i
, , ¯x
n
+ ∆
x
n
) =
∂f
∂x
i
(¯x) + α
i

(∆
x
),
lim
∆
x
α
i
(∆
x
) = 0 ∀i = 1, 2, , n
f(x, y, z)
(x
0
, y
0
, z
0
)
∆
f
=
∂f
∂x
(x
0
, y
0
, z
0

)∆
x
+
∂f
∂y
(x
0
, y
0
, z
0
)∆
y
+
∂f
∂z
(x
0
, y
0
, z
0
)∆
z
+ α∆
x
+ β∆
y
+ γ∆
z

.
ρ =

∆
2
x
+ ∆
2
y
+ ∆
2
z
ǫ =
1
ρ
(α∆
x
+β∆
y
+γ∆
z
)
|ǫ| =
1
ρ
|α∆
x
+ β∆
y
+ γ∆

z
| ≤

(α
2
+ β
2
+ γ
2
)(∆
2
x
+ ∆
2
y
+ ∆
2
z
)

∆
2
x
+ ∆
2
y
+ ∆
2
z
=


α
2
+ β
2
+ γ
2
.
lim
(∆
x
,∆
y
,∆
z
)→0
ǫ = 0
α∆
x
+ β∆
y
+ γ∆
z
= o(ρ).
∆
f
≈
∂f
∂x
(x

0
, y
0
, z
0
)∆
x
+
∂f
∂y
(x
0
, y
0
, z
0
)∆
y
+
∂f
∂z
(x
0
, y
0
, z
0
)∆
z
+ o(ρ).

∆
x
, ∆
y
, ∆
z
∆
f
≈
∂f
∂x
(x
0
, y
0
, z
0
)∆
x
+
∂f
∂y
(x
0
, y
0
, z
0
)∆
y

+
∂f
∂z
(x
0
, y
0
, z
0
)∆
z
.
PTIT
S = arctan
1, 02
0, 95
.
S = arctan
1 + 0, 02
1 − 0, 05
x
0
= 1, y
0
= 1, ∆
x
= 0, 02, ∆
y
= −0, 05 f(x, y) = arctan
x

y
∂f
∂x
=
−y
x
2
+ y
2
,
∂f
∂y
=
x
x
2
+ y
2
.
∆
f
≈
∂f
∂x
(x
0
, y
0
)∆
x

+
∂f
∂y
(x
0
, y
0
)∆
y
.
S = ∆
f
+ f(x
0
, y
0
)
≈
∂f
∂x
(x
0
, y
0
)∆
x
+
∂f
∂y
(x

0
, y
0
)∆
y
+ f(x
0
, y
0
)
= f(1, 1) +
1.0, 02 + 1.0, 05
2
=
π
4
+ 0, 035
= 0, 82rad.
d ∈ R
n
lim
λ→0
f(¯x + λd) −f(¯x)
λ
,
d f ¯x
D
d
f(¯x)
D

d
f(¯x) f(x, y, z) = 2x + 3y + z
2
d = (1, 2, 0), ¯x = (3, −1, 1)
PTIT
D
d
f(¯x)
D
d
f(¯x) = lim
λ→0
f(¯x + λd) − f(¯x)
λ
= lim
λ→0
f(1 + 3λ, −1 + 2λ, 1) −f(3, −1, 1)
λ
= lim
λ→0
2(1 + 3λ) + 3(−1 + 2λ) + 1
2
− (2.3 + 3.(−1) + 1
2
)
λ
= 12.
{e
1
, e

2
, , e
n
} R
n
f(x)
D ¯x ∈ D
D
e
i
f(¯x) =
∂f
∂x
i
(¯x) ∀i = 1, 2, , n.
f : D ⊆ R
n
→ R ¯x ∈ D
d = (d
1
, d
2
, , d
n
)
D
d
f(¯x) =
∂f
∂x

1
(¯x)d
1
+
∂f
∂x
2
(¯x)d
2
+ +
∂f
∂x
n
(¯x)d
n
.
f ¯x
∆
f
= A
1
∆
x
1
+ A
2
∆
x
2
+ + A

n
∆
x
n
+ α
1
∆
x
1
+ α
2
∆
x
2
+ + α
n
∆
x
n
,
∆
f
= f (¯x + ∆
x
) − f(¯x), A
i
=
∂f
∂x
i

(¯x), lim
∆
x
→0
α
i
= 0 i = 1, 2, , n
∆
x
i
= λd
i
D
d
f(¯x) = lim
∆
x
→0
f(¯x + λd) − f(¯x)
λ
= lim
∆
x
→0
(A
1
d
1
+ + A
n

d
n
+ α
1
d
1
+ + α
n
d
n
)
=
∂f
∂x
1
(¯x)d
1
+
∂f
∂x
2
(¯x)d
2
+ +
∂f
∂x
n
(¯x)d
n
.

d = (−1, 3) ¯x = (e, e
2
)
f(x, y) = ln(x
2
+ y).
∂f
∂x
=
2x
x
2
+ y
,
∂f
∂y
=
1
x
2
+ y
.
PTIT
∂f
∂x
(¯x) =
2x
x
2
+ y

(¯x) =
1
e
,
∂f
∂y
(¯x) =
1
x
2
+ y
(¯x) =
1
2e
2
.
D
d
f(¯x) =
∂f
∂x
(¯x)d
1
+
∂f
∂y
(¯x)d
2
=
1

e
(−1) +
1
2e
2
3
=
3
2e
2
−
1
e
.
f : D ⊆ R
n
→ R
m
g : f(D) → R h = gof :
D → R
gof(x) = g

f(x)

g f g
f x = (x
1
, , x
n
) f(x)

h
∂h
∂x
1
=
∂g
∂f
1
∂f
1
∂x
1
+
∂g
∂f
2
∂f
2
∂x
1
+ +
∂g
∂f
m
∂f
m
∂x
1

∂h

∂x
n
=
∂g
∂f
1
∂f
1
∂x
n
+
∂g
∂f
2
∂f
2
∂x
n
+ +
∂g
∂f
m
∂f
m
∂x
n
.
PTIT
∆g
∆

x
1
=
g

f(x + ∆
x
)

− g

f(x)

∆
x
1
=
g

f(x
1
+ ∆
x
1
, , x
n
+ ∆
x
n
)


− g

f(x
1
, , x
n
)

∆
x
1
=
g

f(x
1
+ ∆
x
1
, , x
n
+ ∆
x
n
)

− g

f(x

1
, x
2
+ ∆
x
2
, , x
n
+ ∆
x
n
)

∆
x
1
+ +
g

f(x
1
, , x
n−1
, x
n
+ ∆
x
n
)


− g

f(x
1
, , x
n
)

∆
x
1
=
g

f(x
1
+ ∆
x
1
, )

− g

f(x
1
, x
2
+ ∆
x
2

, )

f(x
1
+ ∆
x
1
, , x
n
+ ∆
x
n
) − f(x
1
, x
2
+ ∆
x
2
, , x
n
+ ∆
x
n
)
×
f(x
1
+ ∆
x

1
, ) − f(x
1
, x
2
+ ∆
x
2
, )
∆
x
1
+ +
g

f(x
1
, , x
n−1
, x
n
+ ∆
x
n
)

− g

f(x
1

, , x
n
)

f(x
1
, , x
n−1
, x
n
+ ∆
x
n
) − f(x
1
, , x
n
)
×
f(x
1
, , x
n−1
, x
n
+ ∆
x
n
) − f(x
1

, , x
n
)
∆
x
1
.
∆
x
1
→ 0
∂h
x
1
=
∂g
f
1
∂f
1
x
1
+
∂g
f
2
∂f
2
x
2

+ +
∂g
f
m
∂f
m
x
1
.
∂f
∂x
f(x, y) = (u
2
+ 1) log
2
v, u = xy, v = 2x + y.
∂f
∂x
=
∂f
∂u
∂u
∂x
+
∂f
∂v
∂v
∂x
= 2u log
2

v.y +
u
2
+ 1
v ln 2
.2
= 2xy
2
log
2
(2x + y) +
2(x
2
y
2
+ 1)
(2x + y) ln 2
.
f : D ⊆ R
n
→ R
∂f
∂x
i
, i = 1, 2, , n
PTIT
∂
∂x
j


∂f
∂x
i

=
∂
2
f
∂x
j
∂x
i
= f”
x
i
x
j
(x), i, j = 1, 2, , n ( i = j)
∂
2
f
∂x
2
j
= f”
x
2
i
(x)
n

f(x, y) = (x
2
+ y
3
) sin 2y.
f
′
x
= 2x sin 2y,
f
′
y
= 3y
2
sin 2y + 2(x
2
+ y
3
) cos 2y.
f
′′
x
2
= 2 sin 2y,
f
′′
xy
= 4x cos 2y,
f
′′

y
2
= 6y sin 2y + 6y
2
cos 2y + 6y
3
cos 2y −4(x
2
+ y
3
) sin 2y
= 2(3y −2x
2
− 2y
3
) sin 2y + y
2
(1 + y) cos 2y.
f : D ⊆ R
2
→ R
M
0
(x
0
, y
0
) ∈ D
∂
2

f
∂x∂y
(M
0
) =
∂
2
f
∂y∂x
(M
0
).
g(x, y) = f (x + ∆
x
, y) −f(x, y),
h(x, y) = f (x, y + ∆
y
) − f(x, y).
g(x, y + ∆
y
) − g(x, y) = h(x + ∆
x
, y) −h(x, y).
g(x, y + ∆
y
) − g(x, y) = g
′
y
(x, y + θ
y

∆
y
).∆
y
0 < θ
y
< 1
= ∆
y

f
′
y
(x + ∆
x
, y + θ
y
∆
y
) − f
′
y
(x, y + θ
y
∆
y
)

= ∆
x

∆
y
f
′′
y x
(x + θ
x
∆
x
, y + θ
y
∆
y
) 0 < θ
x
< 1.
PTIT
h(x + ∆
x
, y) −h(x, y) = ∆
x
∆
y
f
′′
xy
(x + α
x
∆
x

, y + α
y
∆
y
) 0 < α
x
, α
y
< 1.
f
′′
y x
(x + θ
x
∆
x
, y + θ
y
∆
y
) = f
′′
xy
(x + α
x
∆
x
, y + α
y
∆

y
).
(∆
x
, ∆
y
) → 0
f : D ⊆ R
n
→ R
df = f
′
x
1
dx
1
+ f
′
x
2
dx
2
+ + f
′
x
n
dx
n
f df
f d

2
f
d
2
f = d(df ) = d(f
′
x
1
dx
1
+ f
′
x
2
dx
2
+ + f
′
x
n
dx
n
).
n f
d
n
f = d(d
n−1
f).
f(x, y)

d
2
f = f
′′
x
2
dx
2
+ f
′′
xy
dxdy + f
′′
y
2
dy
2
.
f : D ⊆ R
2
→ R f n + 1
M
0
(x
0
, y
0
)
f(x
0

+ ∆
x
0
, y
0
+ ∆
y
0
) − f(x
0
, y
0
) =
1
1!
df(x
0
, y
0
) +
1
2!
d
2
f(x
0
, y
0
) + +
1

n!
d
n
f(x
0
, y
0
)
+
1
(n + 1)!
d
n+1
f(x
0
+ θ ∆
x
, y
0
+ θ ∆
y
).
g(t) = f(x
0
+ t∆
x
0
, y
0
+ t∆

y
0
).
g(t)
g(1) −g(0) = f (x
0
+ ∆
x
0
, y
0
+ ∆
y
0
) − f(x
0
, y
0
).
PTIT
g(t)
g(1) −g(0) =
1
1!
g
′
(0) +
1
2!
g

′′
(0) + +
1
n!
g
(n)
(0) +
1
n!
g
(n+1)
(θ) 0 < θ < 1.
g
′
(0) = f
′
x
(x
0
, y
0
)∆
x
+ f
′
y
(x
0
, y
0

)∆
y
= df(x
0
, y
0
)
g
′′
(0) = f
′′
x
2
(x
0
, y
0
)∆
2
x
+ 2f
′′
xy
(x
0
, y
0
)∆
x
∆

y
+ f
′′
y
2
(x
0
, y
0
)∆
y
2
= df(x
0
, y
0
)

g
(n)
(0) = d
n
f(x
0
, y
0
)
g
(n+1)
(θ) = d

n+1
f(x
0
+ θ ∆
x
, y
0
+ θ ∆
y
).
n = 1
f(x
0
+ ∆
x
0
, y
0
+ ∆
y
0
) − f(x
0
, y
0
) = df(x
0
+ θ ∆
x
, y

0
+ θ ∆
y
) 0 < θ < 1
f(x, y) (x
0
, y
0
)
f(x, y) = 2x
2
− xy −y
2
− 6x − 3y + 5
M
0
(1, −2)
∂f(x, y)
∂x
= 4x − y −6,
∂f(x, y)
∂y
= −x − 2y −3,
∂
2
f(x, y)
∂x
2
= 4,
∂

2
f(x, y)
∂x∂y
= −1,
∂
2
f(x, y)
∂y
2
= −2.
M
0
f(M
0
) = 5,
∂f
∂x
(M
0
) = 0,
∂f
∂y
(M
0
) = 0,
PTIT
∂
2
f
∂x

2
(M
0
) = 4,
∂
2
f
∂x∂y
(M
0
) = −1,
∂
2
f
∂y
2
(M
0
) = −2.
f(x, y) =f (M
0
) +
∂f(M
0
)
∂x
(x − 1) +
∂f(M
0
)

∂y
(y + 2)
+
1
2

∂
2
f(M
0
)
∂x
2
(x − 1)
2
+ 2
∂
2
f(M
0
)
∂x∂y
(x − 1)(y + 2) +
∂
2
f(M
0
)
∂y
2

(y + 2)
2

=5 + 2(x −1)
2
− (x − 1)(y + 2) −(y + 2)
2
.
F : D ⊆ R
2
→ R x, y
F (x, y) = 0.
y = f(x) y = f(x)
x
2
a
2
+
y
2
b
2
= 1,
y = ±
b
a
√
a
2
− x

2
[−a, a]
(x
0
, y
0
) ∈ D F
(i) F (x
0
, y
0
) = 0 (ii) F
′
y
U (x
0
, y
0
)
(iii) F
′
y
(x
0
, y
0
) = 0
y = f(x) V = (x
0
− δ, x

0
+ δ)
f V
f
′
V
f
′
x
= −
F
′
x
F
′
y
(x, y).
y = f(x)
F
′
y
(x
0
, y
0
) = 0
F
′
y
(x

0
, y
0
) > 0.
F
′
y
(x, y) U (x
0
, y
0
) α > 0
F
′
y
(x, y) > 0, ∀(x, y) ∈ [x
0
− α, x
0
+ α] × [y
0
− α, y
0
+ α].
PTIT