已知函数f(x)=alnx-ax-3(a∈R),若函数y=f(x)的图像在点(2,f(2))处的切线的倾斜角为45°,对于任意t∈[1,2],函数g(x)=x^3+x^2[f'(x)+m/2]在区间(t,3)上总不是单调函数,求m的取值范围最好有详细点的
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![已知函数f(x)=alnx-ax-3(a∈R),若函数y=f(x)的图像在点(2,f(2))处的切线的倾斜角为45°,对于任意t∈[1,2],函数g(x)=x^3+x^2[f'(x)+m/2]在区间(t,3)上总不是单调函数,求m的取值范围最好有详细点的](/uploads/image/z/3937880-56-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3Dalnx-ax-3%28a%E2%88%88R%29%2C%E8%8B%A5%E5%87%BD%E6%95%B0y%3Df%28x%29%E7%9A%84%E5%9B%BE%E5%83%8F%E5%9C%A8%E7%82%B9%EF%BC%882%2Cf%EF%BC%882%EF%BC%89%EF%BC%89%E5%A4%84%E7%9A%84%E5%88%87%E7%BA%BF%E7%9A%84%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA45%C2%B0%2C%E5%AF%B9%E4%BA%8E%E4%BB%BB%E6%84%8Ft%E2%88%88%5B1%2C2%5D%2C%E5%87%BD%E6%95%B0g%EF%BC%88x%EF%BC%89%3Dx%5E3%2Bx%5E2%5Bf%27%28x%29%2Bm%2F2%5D%E5%9C%A8%E5%8C%BA%E9%97%B4%EF%BC%88t%2C3%EF%BC%89%E4%B8%8A%E6%80%BB%E4%B8%8D%E6%98%AF%E5%8D%95%E8%B0%83%E5%87%BD%E6%95%B0%2C%E6%B1%82m%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4%E6%9C%80%E5%A5%BD%E6%9C%89%E8%AF%A6%E7%BB%86%E7%82%B9%E7%9A%84)
已知函数f(x)=alnx-ax-3(a∈R),若函数y=f(x)的图像在点(2,f(2))处的切线的倾斜角为45°,对于任意t∈[1,2],函数g(x)=x^3+x^2[f'(x)+m/2]在区间(t,3)上总不是单调函数,求m的取值范围最好有详细点的
已知函数f(x)=alnx-ax-3(a∈R),
若函数y=f(x)的图像在点(2,f(2))处的切线的倾斜角为45°,对于任意t∈[1,2],函数g(x)=x^3+x^2[f'(x)+m/2]在区间(t,3)上总不是单调函数,求m的取值范围
最好有详细点的解答过程。谢谢!
已知函数f(x)=alnx-ax-3(a∈R),若函数y=f(x)的图像在点(2,f(2))处的切线的倾斜角为45°,对于任意t∈[1,2],函数g(x)=x^3+x^2[f'(x)+m/2]在区间(t,3)上总不是单调函数,求m的取值范围最好有详细点的
由(2,f(2))点切线倾斜角为45得,f'(2)=1,即a/2-2=1,则,a=-2,f'(x)=-2/x+2,则g(x)=x^3+x^2(-2/x+2+m/2)=x^3+(2+m/2)x^2-2x,g'(x)=3x^2+(4+m)x-2,题中说函数不单调,也就是说在(t,3)范围内,g'(x)=0有解,因为g'(0)=-20时方程有解,3t^2+(4+m)t-20,解之得-37/3
f'(x)=a/x-a
f'(2)=a/2-a=tan45=1
a=-2
g'(x)=3x^2+2x(-2/x+2+m/2)+2/x^2 * x^2=3x^2+(4+m)x-4
若g(x)在(t,3)上总不是单调函数
1.g'(t)g'(3)<0
参变量分离可得
2.g'(t)g(3)>0
g'(x)对称轴x=(-4-m)/6<0
先求导,求出a,再求m的取值范围。