Re: [考古] 北科94.95微積分考古題
2.
G = x(x^2 + y^2)^1/2 + λ(x^2y-1000√2)
x
Gx= (x^2 + y^2)^1/2 + x──────── + λ(2xy) = 0 ................(1)
(x^2 + y^2)^1/2
y y
Gy= x──────── + λ(x^2) = x(───────── + λx) = 0 .....(2)
(x^2 + y^2)^1/2 (x^2 + y^2)^1/2
y
由(2) x≠0, λ = -───────── 代回(1)
x(x^2 + y^2)^1/2
x y
Gx = (x^2 + y^2)^1/2 + ───────── - ───────── (2xy) = 0
(x^2 + y^2)^1/2 x(x^2 + y^2)^1/2
整理得x^2 + y^2 + x^2 - 2y^2 = 0 → y = ±(√2)x
→ x^2y = ±(√2)x^3 = 1000√2
∴ x = ±10 , y = ±10√2 故 f = f(-10,±10√2) = -10√300
min
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