y=(x+2)^2(x-1)^3 用导数求单调区间

2024-11-15 15:59:18
推荐回答(2个)
回答1:

y'=2(x+2)*(x-1)^3+(x+2)^2 *3(x-1)^2=(x+2)(x-1)^2*(2x-2+3x+6)=(x+2)(5x+4)(x-1)^2
零点为:-2,-4/5,1
在区间(负无穷,-2)及(-4/5,正无穷)上,f'(x)>=0
f(x)单调递增,
同理,在区间(-2,-4/5)上,f'(x)<=0,
f(x)单调减少

回答2:

y=(x+2)^2(x-1)^3
=(x-1+3)^2(x-1)^3
=[(x-1)^2+6(x-1)+9](x-1)^3
=(x-1)^5+6(x-1)^4+9(x-1)^3
y'=5(x-1)^4+24(x-1)^3+27(x-1)^2
=(x-1)^2[(5(x-1)^2+24(x-1)+27]
=(x-1)^2*[5(x-1)+9][(x-1)+3]
=(x-1)^2 *(5x+4)(x+2)
x<-2时,或x>-4/5时,y'>0 单调递增
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