求函数y=(x-1)*x^(2/3)的极值

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求函数y=(x-1)*x^(2/3)的极值求函数y=(x-1)*x^(2/3)的极值求函数y=(x-1)*x^(2/3)的极值其导数为y''=x^(2/3)+(2/3)·x^(-1/3)(x-1)=(5

求函数y=(x-1)*x^(2/3)的极值
求函数y=(x-1)*x^(2/3)的极值

求函数y=(x-1)*x^(2/3)的极值
其导数为y'=x^(2/3)+(2/3)·x^(-1/3)(x-1)=(5/3)·x^(2/3) - (2/3)·x^(-1/3)
令y'=0,则(5/3)·x^(2/3) - (2/3)·x^(-1/3)=0
5·x^(2/3) - 2·x^(-1/3)=0
两边乘x^(1/3)得
5x - 2 =0
x= 2/5
y''=(10/9)·x^(-1/3) + (2/9)·x^(-4/3)
则y''(2/5)恒>0.
说明y(2/5)是极小值,为 (-3/5)·(2/5)^(2/3)= -3·2^(2/3) /5^(5/3)
当x=0时,y''=(10/9)·x^(-1/3) + (2/9)·x^(-4/3)=0,是拐点,不是极值点

答：
y=(x-1)x^(2/3)
求导：
y'(x)=x^(2/3)+(x-1)(2/3)x^(-1/3)
=[x+2x/3-2/3]/x^(1/3)
=(1/3)(5x-2)/x^(1/3)
解y'(x)=0得：x=2/5
当x<0或者x>2/5时，y'(x)>0，y是增函数
当0

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答：
y=(x-1)x^(2/3)
求导：
y'(x)=x^(2/3)+(x-1)(2/3)x^(-1/3)
=[x+2x/3-2/3]/x^(1/3)
=(1/3)(5x-2)/x^(1/3)
解y'(x)=0得：x=2/5
当x<0或者x>2/5时，y'(x)>0，y是增函数
当0所以：
x=0是极大值点，极大值为0
x=2/5是极小值点，极小值为(2/5-1)(2/5)^(2/3)=(-3/25)*20^(1/3)

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