已知函数F(x)=(1/3)x^3-(a/2)x^2+2x=1,且x1,x2是F(x)的两个极值点,0<x1<x2<3(1)求a的取值范围(2)若|x1-x2|≥m^2-2bm-2对b∈[-1,1]恒成立,求实数m的取值范围
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![已知函数F(x)=(1/3)x^3-(a/2)x^2+2x=1,且x1,x2是F(x)的两个极值点,0<x1<x2<3(1)求a的取值范围(2)若|x1-x2|≥m^2-2bm-2对b∈[-1,1]恒成立,求实数m的取值范围](/uploads/image/z/6608050-34-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0F%EF%BC%88x%EF%BC%89%3D%EF%BC%881%2F3%EF%BC%89x%5E3-%EF%BC%88a%2F2%EF%BC%89x%5E2%2B2x%3D1%2C%E4%B8%94x1%2Cx2%E6%98%AFF%EF%BC%88x%EF%BC%89%E7%9A%84%E4%B8%A4%E4%B8%AA%E6%9E%81%E5%80%BC%E7%82%B9%2C0%EF%BC%9Cx1%EF%BC%9Cx2%EF%BC%9C3%EF%BC%881%EF%BC%89%E6%B1%82a%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4%EF%BC%882%29%E8%8B%A5%7Cx1-x2%7C%E2%89%A5m%5E2-2bm-2%E5%AF%B9b%E2%88%88%5B-1%2C1%5D%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82%E5%AE%9E%E6%95%B0m%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
已知函数F(x)=(1/3)x^3-(a/2)x^2+2x=1,且x1,x2是F(x)的两个极值点,0<x1<x2<3(1)求a的取值范围(2)若|x1-x2|≥m^2-2bm-2对b∈[-1,1]恒成立,求实数m的取值范围
已知函数F(x)=(1/3)x^3-(a/2)x^2+2x=1,且x1,x2是F(x)的两个极值点,0<x1<x2<3
(1)求a的取值范围
(2)若|x1-x2|≥m^2-2bm-2对b∈[-1,1]恒成立,求实数m的取值范围
已知函数F(x)=(1/3)x^3-(a/2)x^2+2x=1,且x1,x2是F(x)的两个极值点,0<x1<x2<3(1)求a的取值范围(2)若|x1-x2|≥m^2-2bm-2对b∈[-1,1]恒成立,求实数m的取值范围
(1)f(x)=1\3x^3-a\2x^2+2x+1,f‘(x)=x^2-ax^2+2,x=[a±√(a²-8)]/2,∵0<x1<x2<3,∴0<[a-√(a²-8)]/2<[a+√(a²-8)]/2<3,得2√2<a<11/3.
(2).∵|x1-x2|=√(a²-8)],∴0<|x1-x2|<7/3,
∵|x1-x2|大于等于m^2-2bm-2,∴0<m^2-2bm-2<7/3,当m^2-2bm-2>0时,
{m-[b+√(b²+2)]}{m-[b-√(b²+2)]}>0,b属于-1到1时,[b+√(b²+2)]>0,[b-√(b²+2)]<0,则取
m>[b+√(b²+2)]或m<[b-√(b²+2)];当m^2-2bm-2<7/3时,[b-√(b²+13/3)]<m<[b+√(b²+13/3)],
综合以上,[b-√(b²+13/3)]<m<[b-√(b²+3)];[b+√(b²+3)]m<[b+√(b²+13/3)],结合b属于-1到1,则m范围:-1-4√3/3<m<-1,1<m<1+4√3/3.
第一问用根的分布