Question

设函数y=y(x)由方程2y^3-2y^2+2xy-x^2=1所确定，求函数的极值点并求极值

Answer 1

2y^3-2y^2+2xy-x^2=1
2y^3-y^2-(y^2-2xy+x^2)-1=0
2y^3-y^2-1=(x-y)^2
(y-1)(2y^2+y+1)=(x-y)^2
因(x-y)^2>=0,2y^2+y+1=2(y+1/4)^2+7/8>0
则y-1>=0,y>=1
即y有最小值1,此时x-y=0,x=1
所以函数y=y(x)当x=1时有极小值1

Answer 2

两边对x求导：
6y^2*y'-4y*y'+2y+2xy'-2x=0
即y'=(x-y)/(3y^2-2y+x)
令y'=0,
得：x=y
再将x=y代入原方程，得：2x^3-2x^2+2x^2-x^2=1,
得：2x^3-x^2-1=0
2x^3-2x^2+x^2-1=0
2x^2(x-1)+(x-1)(x+1)=0
(x-1)(2x^2+x+1)=0
得唯一根x=1,
即极值点为(1,
1),
极值即为y=f(1)=1.