已知a>0,函数f(x)=1-axx,x∈(0,+∞).设0<x1<2a,记曲线y=f(x)在点M(x1,f(x1))处的切线为l(1)求l的方程;(2)设l与x轴交点为(x2,0),求证:①0<x2

题目简介

已知a>0,函数f(x)=1-axx,x∈(0,+∞).设0<x1<2a,记曲线y=f(x)在点M(x1,f(x1))处的切线为l(1)求l的方程;(2)设l与x轴交点为(x2,0),求证:①0<x2

题目详情

已知a>0,函数f(x)=
1-ax
x
,x∈(0,+∞).设0<x1
2
a
,记曲线y=f(x)在点M(x1,f(x1))处的切线为l
(1)求l的方程;
(2)设l与x轴交点为(x2,0),求证:①0<x2
1
a
; ②若0<x1
1
a
,则x1<x2<2x1
题型:解答题难度:中档来源:舟山模拟

答案

(1)依题知,得:f′(x)=-class="stub"1
x2
,根据点斜式可得l的方程为y-
1-ax1
x1
=-class="stub"1
x21
(x-x1)

整理得直线l的方程是 class="stub"1
x21
x+y-
2-ax1
x1
=0

(2)证明:由(1)得 x2=x1(2-ax1)
①由于 0<x1<class="stub"2
a
,所以ax1<2,x2=x1(2-ax1)>0
又x2-class="stub"1
a
=x1(2-ax1)-class="stub"1
a
=
a2
x21
-2ax1+1
a
=
(ax1-1)2
a
≤0
,所以,0<x2≤class="stub"1
a

②因为 x2-x1=x1(2-ax1)-x1=x1-ax12=x1(1-ax1),且0<x1<class="stub"1
a
,,所以1-ax1>0,即x1<x2.
又x2-2x1=x1(2-ax1)-2x1=-ax12<0,所以 x2<2x1,
故当0<x1<class="stub"1
a
,则x1<x2<2x1.

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