pdf (it, 2.898,97 KB, 25/01/13)

f : R 2 → R
D := {(x, y) ∈ R 2 : y ≥ |x|}
f(x, y) =
x 2 − y 4
x 2 + sin 2 x + y .
lim f(x, y), lim
(x,y)→(0,0)
(x,y)→(0,0)
(x,y)∈D
f(x, y)
(t, 0) t → 0 ±
lim f(t, 0) = lim
t→0 ± t→0 ±
t2 t2 + sin2 1
=
t 2 .
(0, t) t → 0 ±
lim f(0, t) = lim
t→0 ± t→0 ±
−t4 t
(x, y) ∈ D f(x, y) ≤ 0
0
f(x, y) ≥
−y 4
x 2 + sin 2 x + y
= 0.
≥ −y3
F (x, y) = (x 2 + y 2 ) 2 − (x 2 − y 2 ),
¯ B = {(x, y) ∈ R 2 : x 2 + y 2 ≤ 1}
∂xF (x, y) = 4x x 2 + y 2 − 2x
∂yF (x, y) = 4y x 2 + y 2 + 2y.
⎧
⎪⎨ ∂xxF (x, y) = −2 + 12x2 + 4y2 ∂yyF (x, y) = 2 + 4x 2 + 12y 2
⎪⎩
∂xyF (x, y) = 8xy
−2 2
∂B = {(x, y) : x 2 + y 2 = 1}