求函数的极值：f(x,y)=x^3+8y^3-6xy+5

题目

求函数的极值：f(x,y)=x^3+8y^3-6xy+5
答案是这个：f(1,1/2)=4
问是怎么得出来的！是用求导作

答案

df/dx = 3 x^2 - 6 y
df/dy = -6 x + 24 y^2
令df/dx = df/dy = 0
解得
(1) x = 0,y = 0
(2) x = 1,y = 1/2
d²f/dx² = 6x
d²f/dxdy = d²f/dydx = -6
d²f/dy² = 48y
det[d²f/dx²,d²f/dxdy; d²f/dydx,d²f/dy²]
= d²f/dx² · d²f/dy² - d²f/dxdy · d²f/dydx
= 288·x·y - 36
(1) x = 0,y = 0,上式 = -36 < 0
(0,0)为极大值点,f(0,0) = 5
(2) x = 1,y = 1/2,上式 = 108 > 0
(1,1/2)为极小值点,f(1,1/2) = 4