Question

设函数f(x)=(x-1)(x-2)(x-3)(x-4),则f'(x)=0有

A,分别位于区间(1,2),(2,3),(3,4)内的三个根
B，四个根xi(i=1,2,3,4)
C，分别位于区间(0,1),(1,2),(2,3),(3,4)内四个根
D,分别位于区间(0,1),(1,2),(2,3)内三个根

Answer 1

1、因为f(x)=(x-1)(x-2)(x-3)(x-4)的根是x=1,2,3,4
画图易知f'(x)=0的三个根,分别位于区间(1,2),(2,3),(3,4)内
2、
lny=ln(x-1)(x-2)(x-3)(x-4)=ln(x-1)+ln(x-2)+ln(x-3)+ln(x-4)
(lny)'=(1/y) *y'=1/(x-1 )+1/(x-2)+1/(x-3)+1/(x-4)
f'(x)=y'={1/(x-1 )+1/(x-2)+1/(x-3)+1/(x-4)}*{(x-1)(x-2)(x-3)(x-4)}
=(x-2)(x-3)(x-4)+(x-1)(x-3)(x-4)+(x-1)(x-2)(x-4)+(x-1)(x-2)(x-3)
该函数是三次有三个根 排除B、C
然后判断f'(0)f1(1)的符号都小于0 所以选A

追问

f(X)的导数函数是什么？怎样画图？

追答

f(X)的导数函数是(x-2)(x-3)(x-4)+(x-1)(x-3)(x-4)+(x-1)(x-2)(x-4)+(x-1)(x-2)(x-3)=4x^3-30x^2+70x-50
画图只需简单画图函数f(x)的图像 或者想象一下f(x)=(x-1)(x-2)(x-3)(x-4)的根是x=1,2,3,4
于是在区间(1,2),(2,3),(3,4)内有极值(上凸或者下凸)这时候的倒数值就是0

Answer 2

A