Question

已知a b c属于R 且三次方程f(X)=X三次方-aX平方+bX-c有三个实根x1 x2 x3.

Answer 1

1> 因为x1 x2 x3为根,

=>x^3-a*x^2+b*x-c=(x-x1)*(x-x2)*(x-x3)

右边展开为

x^3-(x1+x2+x3)*x^2+(x1*x2+x1*x3+x2*x3)*x-x1*x2*x3

=>a=x1+x2+x3,b=x1*x2+x1*x3+x2*x3,

c=x1*x2*x3

2> 由 c=x1*x2*x3 >0 知只能有偶数个根为负

不妨设x1,x20

=>a*x2=x1*x2+x1*x3+x2^2 0 (2式)

(-1)*1式+(2式) =>x1*x3-x2^2>0

然而 x1*x30 矛盾,

=>x1 x2 x3都大于0

3> 由f(x)在x=m、n处取极值知

f'(x)=3*x^2-2a*x+b=0 的根为m,n

又-1