求函数f(x)=(x-1)²(x+1)³的单调区间与极ŀ
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f(x)=(x-1)^2
*
(x+1)^3
f'(x)
=
(x-1)^2
*
3(x+1)^2
+
(x+1)^3
*
2(x-1)
=
(x+1)^2
*
(x-1)
*
{
3(x-1)+2(x+1)
}
=
(x+1)^2
*
(x-1)
*
(5x-1)
=
5(x+1)^2
*
(x-1/5)
*
(x-1)
x<1/5或x>1时,f'(x)>0,f(x)单调增;1/5<x<1时,f'(x)<0,f(x)单调减少
∴单调增区间:(-无穷大,1/5)u(1,+无穷大);单调减区间(1/5,1)
x=1/5时有极大值f(1/5)
=
(1/5-1)^2
*
(1/5+1)^3
=
3456/3125
x=1时有极小值f(1)
=
0
*
(x+1)^3
f'(x)
=
(x-1)^2
*
3(x+1)^2
+
(x+1)^3
*
2(x-1)
=
(x+1)^2
*
(x-1)
*
{
3(x-1)+2(x+1)
}
=
(x+1)^2
*
(x-1)
*
(5x-1)
=
5(x+1)^2
*
(x-1/5)
*
(x-1)
x<1/5或x>1时,f'(x)>0,f(x)单调增;1/5<x<1时,f'(x)<0,f(x)单调减少
∴单调增区间:(-无穷大,1/5)u(1,+无穷大);单调减区间(1/5,1)
x=1/5时有极大值f(1/5)
=
(1/5-1)^2
*
(1/5+1)^3
=
3456/3125
x=1时有极小值f(1)
=
0
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