x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

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1 2011 I 2 II III 17, 18, : : : x = a x = a

2 x = a 1 f (a r, a + r) f(a) r a f f(a) 2 2. (a, b) 2 f (a, b) r f(a, b) r (a, b) f f(a, b)

3 2 3 (a, b) f(a, b) f f 1 x = a f f (a) = 0. x = a (a r, a + r) f(a) f(x) f(x) f(a) 0 f(x) f(a) a r < x < a = 0 x a f(x) f(a) a < x < a + r = 0 x a f x a f (a) f (a) = x a 0 x a+0 f(x) f(a) 0 x a f(x) f(a) 0 x a f (a) = 0 0 x 2 xy 2 f (a, b) f f (a, b) = (a, b) = 0 x y. f (a, b) y b x 1 f(x, b) x = a f(x, b) x x = a 0 f x (a, b) = 0 f y (a, b) = 0 xy 0 f(x) = x 3 0

4 : 1 1 f (a) = 0 a 2 f (a) > 0 f (a) < 0 f (a) = 0 f(x) = x 4, x 4, x 3 x = 0 f 0 f 0 2 f (a) = 0 f (a) > 0 f C 2 f f (a) = (f ) (a) > 0 f (x) a f (a) = 0 a r < x < a f (x) < 0 f(x) a < x < a + r f (x) > 0 f(x) x = a f f > 0 f f (a) > 0 f (x) a f (a) = 0 1 a

5 x = a f(x) x = a f (a) = f(a) + f (a) (x a) 2 2 f (a) > 0 x = a f (a) < 0 x = a x = a 2 f(x) f (a) > 0 x = a f (a) < 0 x = a f(x) = f(a) + f (a)(x a) f (a + θ(x a))(x a) 2 θ 0 1 a f (a) = 0 f(x) = f(a) f (a + θ(x a))(x a) 2 x a f f C 2 f (a) f (a + θ(x a)) f (a) > 0 f(x) = f(a) f (a + θ(x a))(x a) 2 > f(a) f(a) f (a) < 0 f(x) = f(a) f (a + θ(x a))(x a) 2 < f(a) f(a) 1.3 : f(x, y) (a, b) f x (a, b) = f y (a, b) = 0 f x (a, b) = f y (a, b) = 0 f (a, b) n n

6 f(x, y) f f (a, b) = (a, b) = 0 x y (a, b) f f(x, y) = x 2 + y 3 (0, 0) f(0, y) y y (0, 0) f(0, 0) = f (a) = P (x) = f(a) + p(x a) 1 f(x) x = a 1 f(x) P (x) = 0 x a x a p = f (a) P (x, y) = f(a, b) + p(x a) + q(x a) 2 f(x, y) (a, b) 1 (x,y) (a,b) f(x, y) P (x, y) (x a)2 + (y b) 2 = P (x, y) p q p = f (a, b), x q = f (a, b) y 1 2 Q(x) 1 f(x) x = a 2 f(x) Q(x) x a (x a) 2 = 0

7 2 7 Q(x) Q(x) = f(a) + f (a)(x a) + f (a) (x a) Q(x, y) f(x, y) (a, b) 2 (x,y) (a,b) f(x, y) Q(x, y) = 0 (1) (x a)2 + (y b) 22 (1) Q(x, y) 2 (x, y) (a, b) (a, b) x = a + r cos θ, y = b + r sin θ (x, y) (a, b) r 0 r 0 θ r 0 r 0 θ (1) Q(x, y) x a y b 2 c + p(x a) + q(y b) + A(x a) 2 + 2B(x a)(y b) + C(y b) 2 c, p, q, A, B, C (1) x = a + r cos θ, y = b + r sin θ r 0 1 r 2 { f(a + r cos θ,b + r sin θ) c pr cos θ qr sin θ Ar 2 cos 2 θ 2Br 2 cos θ sin θ Cr 2 sin 2 θ } = 0 (2) r 0 r 2 0 r 2 0 r = 0 0 f(x, y) c = f(a, b) r 2 r r 0 r 2 r r 0 r 2 0 f(a + r cos θ, b + r sin θ) f(a, b) pr cos θ qr sin θ = 0 r 0 r x, y f(x, y) f(a, b) p(x a) q(y b) = 0 (x,y) (a,b) (x a)2 + (y b) 2 (a, b) p = f (a, b), x q = f (a, b) y

8 2 8 A, B, C 2 (2) r 1 θ (2) h(r) r 2 g(r) θ h(0) = g(0) = 0 h(r) h(r) h(0) = g(r) g(r) g(0) = h (ρ) g (ρ) ρ 0 r ρ r θ g (ρ) = 2ρ h (ρ) h (ρ) = f f (a + ρ cos θ, b + ρ sin θ) cos θ + (a + ρ cos θ, b + ρ sin θ) sin θ x y f f (a, b) cos θ x y (a, b) sin θ 2Aρ cos2 θ 4Bρ cos θ sin θ 2Cρ sin 2 θ h (r) r 0 g (r) = 0 (f x (a + r cos θ, b + r sin θ) f x (a, b)) cos θ + (f y (a + r cos θ, b + r sin θ) f y (a, b)) sin θ r 0 2r = A cos 2 θ + 2B cos θ sin θ + C sin 2 θ (3) θ (2) (3) (2) f x f y φ(r) = f x (a + r cos θ, b + r sin θ), ψ(r) = f y (a + r cos θ, b + r sin θ) 1 f x (a + r cos θ, b + r sin θ) f x (a, b) = φ (r) = f x r 0 r x (a, b) cos θ + f x (a, b) sin θ y f y (a + r cos θ, b + r sin θ) f y (a, b) = ψ (r) = f y r 0 r x (a, b) cos θ + f y (a, b) sin θ y (4) φ(r) ψ(r) (4) (3) f 2 x 2 (a, b) cos2 θ + 1 ( 2 ) f 2 y x (a, b) + 2 f (a, b) cos θ sin θ f x y 2 y 2 (a, b) sin2 θ = A cos 2 θ + 2B cos θ sin θ + C sin 2 θ θ θ = 0 θ = π/2 A = 1 2 f 2 x 2 (a, b) C = 1 2 f (a, b) 2 y2

9 2 9 2B = 1 2 ( 2 ) f y x (a, b) + 2 f (a, b) x y f(x, y) (a, b) f x (x, y) f y (x, y) (a, b) (a, b) f(x, y) 2 Q(x, y) Q(x, y) =f(a, b) + f f (a, b)(x a) + x y (a, b)(y b) ( 2 ) f 2 y x (a, b) + 2 f (a, b) (x a)(y b) + 1 x y 2 2 f (a, b)(x a)2 x2 2 f (a, b)(y b)2 y2 f, f x, f y C 1 C 1 f x f y f f x f y C 1 f C 2 C C 2 4 C 2 2 f xy = f yx (x a)(y b) : C 2 2 f(x, y) C 2 f (a, b) 2 2 f(a, b) + f f (a, b)(x a) + (a, b)(y b) x y f 2 x 2 (a, b)(x a)2 + 2 f x y (a, b)(x a)(y b) f (a, b)(y b)2 2 y2

10 (a, b) 2 f(x, y) 2 f 2 f 1 1 f(x) x = a f (a) = 0 x = a 2 f(a) + f (a) (x a) 2 2 f (a) > 0 f f (a) < 0 f 2 (a, b) f f x (a, b) = f y (a, b) = 0 f (a, b) 2 f(a, b) x 2 (a, b)(x a)2 + 2 f y x (a, b)(x a)(y b) f (y b)2 2 y2 f 2 (a, b) f(a, b) 2 1/2 u = x a, v = y b u v 2 (0, 0) A = 2 f (a, b) x2 B = 2 f (a, b) y x C 2 Q(u, v) Q(u, v) = Au 2 + 2Buv + Cv 2 = 2 f (a, b) y2 z = Q(u, v) (0, 0, 0) u v u v 2 A, B, C 0 Q(u, v) 0

11 2 11 A, B, C 0 A 0 Q(x, y) = A ( ( u + B ) ) 2 A v AC B2 + A 2 v 2 1 v = u 2 v = 2u 2 z = Q(x, y) A A z = (u + BA ) 2 v + AC B2 A 2 v 2 z = (u + BA ) 2 v AC B2 A 2 v 2 2 u v 1 X, Y X = u + B A v AC B 2 AC B 2 Y = v AC B 2 A 2 B2 AC Y = v AC B 2 = 0 Y = v A 2 A > 0, AC B 2 > 0 z = X 2 + Y 2 A > 0, AC B 2 = 0 z = X 2 A > 0, AC B 2 < 0 z = X 2 Y 2 A < 0, AC B 2 > 0 z = X 2 Y 2 A < 0, AC B 2 = 0 z = X 2 A < 0, AC B 2 < 0 z = X 2 + Y 2 A 0 A = 0 C 0 x y A = C = 0 B 0 2Buv = B 2 ( (u + v) 2 (u v) 2) 2 z = Q(u, v) uv

12 2 12 B z = X 2 Y 2 z = X 2 + Y 2 A 0 z = X 2 Y 1 Y z = X 2 z = X 2 1 z = X 2 z = X 2 1: 1 2 z = X 2 + Y 2 XY (x, y) = (0, 0) XY z = k X 2 + Y 2 = k k z = X 2 + Y 2 1 z 1 Xz Xz Y = 0 x = X 2 + Y 2 z = X 2 z z = X 2 Y 2 2 X, Y u, v uv z = X 2 Y 2 z = X 2 + Y 2 X Y z = X 2 Y 2 XY X 2 Y 2 = k k X k Y k = 0 Y = ±X 3

13 2 13 z = X 2 + Y 2 z = X 2 Y 2 2: Y z = 0 z = 2 O X z = 2 z = 3 z = 1 z = 3 z = 1 3: z = X 2 Y 2 XY Xz Y = k z = X 2 k 2 X = 0 k z = k 2 Y z Xz z = X 2 Y z z = Y 2 z = X 2 Y 2 4 z = f(x, y) (a, b) 2 z = Q(u, v) (0, 0) z = Q(u, v) (a, b) z = X 2 + Y 2 X 2 + Y 2 > 0 (0, 0) (a, b) z = X 2 Y 2 (X, Y ) (0, 0) X 2 Y 2 < 0 (0, 0)

14 2 14 z = X 2 Y 2 4: z = X 2 Y 2 (a, b) z = X 2 Y 2 (X, Y ) (0, 0) X > Y X 2 Y 2 > 0 X < Y X 2 Y 2 < 0 (0, 0) z = X 2 Y AC B 2 = 0 2 z = X 2 2 (0, 0) (0, 0) (0, Y ) 0 2 (0, Y ) (0, 0) (a, b) 2 (0, 0) 2 f(x, y) = x 2 + y 4, g(x, y) = x 2 + y 3 (0, 0) 2 x 2 f g (a, b) C 2 f(x, y) A = 2 f (a, b), x2 B = 2 f (a, b), y x C = 2 f (a, b) y2 (1) AC B 2 > 0 A > 0 C > 0 (a, b) (2) AC B 2 > 0 A < 0 C < 0 (a, b) (3) AC B 2 < 0 (a, b) (4) AC B 2 = 0 2 (a, b)

15 : 2 (a, b) (x, y) f(x, y) φ(x) = φ(a) + φ (a)(x a) + φ (a + θ(x a)) (x a) 2 0 < θ < 1 2 (a, b) f (x 0, y 0 ) (a, b) (x 0, y 0 ) 2 ( x y ) = ( a b ) + t ( x 0 a y 0 b t f(x, y) φ(t) φ(t) = f(a + t(x 0 a), b + t(y 0 b)) ) φ(0) = f(a, b), φ(1) = f(x 0, y 0 ) φ(t) t = 1, a = 0 φ(1) = φ(0) + φ (0)1 + φ (θ 1) 1 2 = φ(0) + φ (0) + φ (θ) 2 2 (5) θ (0, 1) φ (t) φ (t) = f x (a + t(x 0 a),b + t(y 0 b))(x 0 a) + f y (a + t(x 0 a), b + t(y 0 b))(y 0 b) φ (0) = f x (a, b)(x 0 a) + f y (a, b)(y 0 b) f C 2 φ (t) f xy = f yx φ (t) = 2 f x 2 (a + t(x 0 a), b + t(y 0 b))(x 0 a) f y x (a + t(x 0 a), b + t(y 0 b))(x 0 a)(y 0 b) + 2 f y 2 (a + t(x 0 a), b + t(y 0 b))(y 0 b) 2

16 (5) (x 0, y 0 ) (x, y) 2 f(x, y) = f(a, b)+ f f (a, b)(x a) + (a, b)(y b) x y f (a + θ(x a), b + θ(y b))(x a)2 2 x2 + 2 f (a + θ(x a), b + θ(y b))(x a)(y b) y x f (a + θ(x a), b + θ(y b))(y b)2 2 y2 (4) f(x, y) = x 2 + y 4 g(x, y) = x 2 + y 3 (1)(2)(3) (1) (a, b) C 2 f(x, y) A = 2 f (a, b), x2 B = 2 f (a, b), y x C = 2 f (a, b) y2 AC B 2 > 0 A > 0 (a, b) (x, y) θ A = 2 f (a + θ(x a), b + θ(y b)) x2 B = 2 f (a + θ(x a), b + θ(y b)) y x C = 2 f (a + θ(x a), b + θ(y b)) y2 (a, b) f x (a, b) = f y (a, b) = 0 f(x, y) = f(a, b) A (x a) 2 + B (x a)(y b) C (y b) 2 u = x a, v = y b 2 2 f C 2 f xx f xx (a, b) = A > 0 (x, y) (a, b) r 1 (a, b) r 1 (x, y) f xx (a + θ(x a), b + θ(y b)) = A > 0 A ( ) 2 A u 2 + 2B uv + C v 2 = A u + B A v + A C B 2 A v 2

17 2 17 f C 2 g(x, y) g(x, y) = f xx (x, y)f yy (x, y) (f xy (x, y)) 2 g(x, y) g(a, b) = f xx (a, b)f yy (a, b) (f xy (a, b)) 2 = AC B 2 > 0 (x, y) (a, b) r 2 (a, b) r 2 (x, y) g(a + θ(x a), b + θ(y b)) = A C B 2 > 0 A u 2 + 2B uv + C v 2 = ( A )) 2 (u + B A v + A C B 2 v (u, v) (0, 0) 2 (x, y) (a, b) (x, y) (a, b) r 1 r 2 r (a, b) r (x, y) f(x, y) = f(a, b) + > f(a, b) f(a, b) A (1) x2 9 y2 (2) 2x 2 2xy + 5y 2 6x + 12y (3) x 2 + xy + y x + y (4) x 3 9xy + y xy (5) xy(x 2 + y 2 1) (6) (x + y)e xy (7) (x 2 + y 2 )e x2 y (f xy ) 2 f xx f yy = 0 2 2

18 f(x, y) = (y x 2 )(y 2x 2 ) f f x (x, y) = 2x(3y 4x 2 ), f y (x, y) = 2y 3x 2 (0, 0) 2 f xx (x, y) = 6(4x 2 y), f xy (x, y) = 6x, f yy (x, y) = 2 (0, 0) f xx (0, 0) = f xy (0, 0) = 0, f yy (0, 0) = 2 f xx (0, 0)f yy (0, 0) (f xy (0, 0)) 2 = 0 (0, 0) z = f(x, y) f(0, 0) = 0 f(x, y) = 0 (x, y) f(x, y) = (y x 2 )(y 2x 2 ) = 0 y = x 2 y = 2x 2 f 0 x 2 < y < 2x 2 f(x, y) < 0 y < x 2 y > 2x 2 f(x, y) > 0 (0, 0) f(x, y) f(0, 0) = 0 y O x 5: f(x, y) y = x 2 y = 2x 2 0 f(x, y) (0, 0) (0, 0)

19 2 19 (5) x y 0 2. f(x, y) (a, b) z = f(x, y) (a, b) z (a, b) (a, b) x = at, y = bt f(x, y) f(at, bt) = (bt a 2 t 2 )(bt 2a 2 t 2 ) = t 2 (b a 2 t)(b 2a 2 t) t = 0 a, b f(at, bt) b > 0 b/2a 2 < t < b/2a 2 (b a 2 t)(b 2a 2 t) > 0 b = 0 f(at, 0) = 2a 4 t 4

20 (1) X = x/3, Y = y/5 X 2 Y 2 (0, 0) (2) f f x (x, y) = 4x 2y 6 f y (x, y) = 2x + 10y + 12 (1, 1) f x 1 y + 1 f(x, y) = 2(x 1) 2 2(x 1)(y + 1) + 5(y + 1) = 9 > 0, 2 > 0 (1, 1) 7 2 (3) f f x (x, y) = 2x + y 3 x 2 f y (x, y) = x + 2y 3 y 2 (1, 1) 2 f xx (x, y) = x 3 f xy (x, y) = 1 f yy (x, y) = y 3 f xx (1, 1) = f yy (1, 1) = 8, f xy (1, 1) = f yx (1, 1) = 1 f xx (1, 1)f yy (1, 1) f xy (1, 1) 2 = 63 > 0 f xx (1, 1) > 0 (1, 1) 9 (4) f f x (x, y) = 3x 2 9y f y (x, y) = 9x + 3y 2 (0, 0) (3, 3) 2 f xx (x, y) = 6x f xy (x, y) = 9 f xy (x, y) = 6y f xx (0, 0) = f yy (0, 0) = 0, f xy (0, 0) = f yx (0, 0) = 9 f xx (0, 0)f yy (0, 0) f xy (0, 0) 2 = 81 < 0 (0, 0) f xx (3, 3) = f yy (3, 3) = 18, f xy (3, 3) = f yx (3, 3) = 9 f xx (3, 3)f yy (3, 3) f xy (3, 3) 2 = 243 > 0 f xx (3, 3) > 0 (3, 3) 0

21 2 21 (5) f f x (x, y) = y(3x 2 + y 2 1) f y (x, y) = x(x 2 + 3y 2 1) (0, 0), (±1, 0), (0, ±1), (±1/2, ±1/2) 9 2 f xx (x, y) = 6xy f xy (x, y) = 3x 2 + 3y 2 1 f yy (x, y) = 6xy f xx (0, 0) = f yy (0, 0) = 0, f xy (0, 0) = f yx (0, 0) = 1 f xx f yy fxy 2 = 1 < 0 (0, 0) (±1, 0), (0, ±1) 4 f xx = f yy = 0, f xy = f yx = 2 f xx f yy fxy 2 = 4 < 0 ±(1/2, 1/2) f xx = f yy = 3/2, f xy = f yx = 1/2 f xx f yy fxy 2 = 2 > 0 f xx > 0 1/8 ±(1/2, 1/2) f xx = f yy = 3/2, f xy = f yx = 1/2 f xx f yy fxy 2 = 2 > 0 f xx < 0 1/8 (5) 9 f(x, y) 1 x, y 2 x 2 + y 2 1 f(x, y) = 0 x y f(x, y) 6 (0, 0), (±1, 0), (0, ±1) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1 2, 1 2 ) ( 1 2, 1 2 ) (6) f f x (x, y) = (1 xy y 2 )e xy f y (x, y) = (1 x 2 xy)e xy ±(1/ 2, 1/ 2) 2 f xx (x, y) = (xy 2 + y 3 2y)e xy f xy (x, y) = (x 2 y + xy 2 2x 2y)e xy f yy (x, y) = (x 3 + x 2 y 2x)e xy

22 2 22 y 0 x 6: (5) 2 f xx = f yy = 1/ 2e, f xy = f yx = 3/ 2e f xx f yy fxy 2 = 1 2e 9 2e = 4 e < 0 (7) f f x (x, y) = 2x(1 + x 2 + y 2 )e x2 y 2 f y (x, y) = 2y(1 x 2 y 2 )e x2 y 2 (0, 0), (0, ±1) 2 f xx (x, y) = 2(1 + 3x 2 + y 2 + 2x 4 + 2x 2 y 2 )e x2 y 2 f xy (x, y) = 4xy(x 2 + y 2 )e x2 y 2 f yy (x, y) = 2(1 x 2 3y 2 + 2x 2 y 2 + 2y 4 )e x2 y 2 f xx (0, 0) = f yy (0, 0) = 2, f xy (0, 0) = f yx (0, 0) = 0 f xx (0, 0)f yy (0, 0) f xy (0, 0) 2 = 4 > 0 f xx (0, 0) > 0 (0, 0) 0 (0, ±1) f xx = 4/e, f xy = f yx = 0, f yy = 4/e f xx f yy f 2 xy = 16 e 2 < 0

40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x,

40 6 y mx x, y 0, 0 x 0. x,y 0,0 y x + y x 0 mx x + mx m + m m 7 sin y x, x x sin y x x. x sin y x,y 0,0 x 0. 8 x r cos θ y r sin θ x, y 0, 0, r 0. x, 9.. x + y + 0. x,y, x,y, x r cos θ y r sin θ xy x y x,y 0,0 4. x, y 0, 0, r 0. xy x + y r 0 r cos θ sin θ r cos θ sin θ θ 4 y mx x, y 0, 0 x 0. x,y 0,0 x x + y x 0 x x + mx + m m x r cos θ 5 x, y 0, 0,

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e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,, 01 10 18 ( ) 1 6 6 1 8 8 1 6 1 0 0 0 0 1 Table 1: 10 0 8 180 1 1 1. ( : 60 60 ) : 1. 1 e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1,

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