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(2kπ, π/2+2hπ) (π+2kπ, 3π/2+2hπ) h, k ∈ Z
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(2kπ, 3π/2+2hπ) (π +2kπ, π/2+2hπ) h, k ∈ Z
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(π/2 + kπ, hπ) h, k ∈ Z
∂xf(x, y) = 4x 3 + 2xy = x(4x 2 + y), ∂yf(x, y) = x 2 + 2y.
(0, 0)
∂ 2 xxf(x, y) = 12x 2 + 2y ∂ 2 yyf(x, y) = 2 ∂ 2 xyf(x, y) = 2x
D 2 f(0, 0) =:
0 0
0 2
.
(0, 0)
f(x, y) = g(x 2 , y) g(v, w) = v 2 + vw + w 2 + 3
v 2 + vw + w 2 v > 0 v = x 2 > 0 x = 0 v > 0
v 2 + vw + w 2 = 0 w
v 2 − 4v 2 < 0 v 2 + vw + w 2 v > 0
w → ±∞ v > 0
f(x, y) = g(x 2 , y) > 3 = f(0, 0) x = 0 (0, 0)
f(x, y) = x 4 + y 4 − 2(x 2 + y 4 ) + 4xy
∂xf(x, y) = 4x 3 − 4x + 4y, ∂yf(x, y) = −4y 3 + 4x.
x = y 3 4y 9 − 4y 3 + 4y = 0 x = y 3
y(y 8 −y 2 +1) = 0 (0, 0)
y 8 −y 2 +1 = 0 y = 0 0 < |y| ≤ 1 1−y 2 ≥ 0 y 8 −y 2 +1 ≥ y 8 > 0
|y| > 1 y 8 > y 2 y 8 − y 2 + 1 > 1 > 0
∂ 2 xxf(x, y) = 12x 2 − 4 ∂ 2 yyf(x, y) = −12y 2 ∂ 2 xyf(x, y) = 4
D 2 f(0, 0) =:
−4 4
4 0
,
λ 2 + 4λ − 16 = 0 λ = −2 ± 2 √ 5