Question

已知函数f(x)=(x-1)/(x+1),x∈[1,3],求函数的最大值和最小值

已知函数f(x)=(x-1)/(x+1),x∈[1,3],求函数的最大值和最小值

Answer 1

f(x)=(x+1-2)/(x+1)
=(x+1)/(x+1)-2/(x+1)
=1-2/(x+1)

1<=x<=3
2<=x+1<=4
所以1/4<=1/(x+1)<=1/2
-1<=-2/(x+1)<=-1/2
1-1<=1-2/(x+1)<=1-1/2
0<=f(x)<=1/2
所以最大值=1/2，最小值=0

Answer 2

f'(x)=2/(x+1)^2
x∈[1,3]
此时，f'(x)>0
故，f（x）为增函数，且是单调的
最小值f(1)=0
最大值f(3)=1/2

Answer 3

f(x)=(x-1)/(x+1)=1-2/(x+1)
x∈[1,3],(x+1)∈[2,4],2/(x+1)∈[1/2,1],-2/(x+1)∈[-1,-1/2],
f(x)∈[0,1/2].

Answer 4

8 0