作业帮f(x)=(x-2006)(x-2007)(x-2008)(x-2009)最小值?
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/27 17:36:10
f(x)=(x-2006)(x-2007)(x-2008)(x-2009)最小值?
f(x)=(x-2006)(x-2007)(x-2008)(x-2009)最小值?
f(x)=(x-2006)(x-2007)(x-2008)(x-2009)最小值?
f(x)=(x-2006)(x-2007)(x-2008)(x-2009)
设a=x-2009
则(x-2006)(x-2007)(x-2008)(x-2009)
=a(a+1)(a+2)(a+3)
=(a^2+3a)(a^2+3a+2)
=(a^2+3a)^2+2(a^2+3a)
=(a^2+3a+1)^2-1
=[(a+3/2)^2-5/4]^2-1
因为[(a+3/2)^2-5/4]^2≥0,所以[(a+3/2)^2-5/4]^2-1≥-1
所以f(x)=(x-2006)(x-2007)(x-2008)(x-2009)最小值是-1
x=2006.5,f(x)=-15/16
f(x)=[(x-2006)(x-2009)][(x-2007)(x-2008)]
=[(x²-4015)+4030054][(x²-4015)+4030056]
=(x²-4015)²+8060110(x²-4015)+16241343303024
令a=x²-4015x
=(x-4015/2)²...
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f(x)=[(x-2006)(x-2009)][(x-2007)(x-2008)]
=[(x²-4015)+4030054][(x²-4015)+4030056]
=(x²-4015)²+8060110(x²-4015)+16241343303024
令a=x²-4015x
=(x-4015/2)²-16120225/4>=-16120225/4
y=f(x)=a²+8060110a+16241343303024
=(a+4030056)²-1
a>=-16120225/4=-4030056.25
对称轴x=-4030056
所以x=-4030056，y最小=-1
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