【题文】函数y=ax在[0,1]上的最大值与最小值之和为3,则函数y=3ax-1在[0,1]上的最大值与最小值的差是A.6B.1C.3D.
题型:难度:来源:
【题文】函数y=a
x在[0,1]上的最大值与最小值之和为3,则函数y=3a
x-1在[0,1]上的最大值与最小值的差是
A.6 | B.1 | C.3 | D.![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010503-35751.png) |
答案
【答案】
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010503-82482.png)
解析
【解析】
试题分析:(1)若
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-23116.png)
时,指数函数y=a
x在[0,1]上为增函数,当
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-81838.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-38956.png)
取最小值为1,当
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010505-89770.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-38956.png)
取最大值为
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010506-40832.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010506-79138.png)
;(2)若
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010506-48081.png)
时,指数函数y=a
x在[0,1]上为减函数,当
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-81838.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-38956.png)
取最大值为1,当
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010505-89770.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-38956.png)
取最小值为
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010506-40832.png)
,则
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010506-79138.png)
,不满足
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010506-48081.png)
;综合(1)(2)所述:
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010507-60416.png)
. 而y=3a
x-1![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010507-34513.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010507-79972.png)
上是单调增函数,当
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-81838.png)
时,
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-38956.png)
取最小值
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010508-39836.png)
,当
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010505-89770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010504-38956.png)
取得最大值为
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010508-14293.png)
,最大值与最小值的差是
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010508-39836.png)
考点:1.指数函数的单调性与最值
举一反三
【题文】(本题满分12分)
(1)
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010443-40649.png)
;
(2)
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010444-26999.png)
.
【题文】(本题满分12分)已知函数
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010427-96532.png)
,其中常数a,b为实数.
(1)当a>0,b>0时,判断并证明函数
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010427-99545.png)
的单调性;
(2)当ab<0时,求
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010428-51360.png)
时的
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010428-23120.png)
的取值范围.
【题文】(本题满分15分)已知
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010411-91451.png)
,
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010411-45260.png)
(1)若f(x)的最小值记为h(a),求h(a)的解析式.
(2)是否存在实数m,n同时满足以下条件:①
![](http://img.shitiku.com.cn/uploads/allimg/20200328/20200328010412-81113.png)
;②当h(a)的定义域为[n,m]时,值域为[n
2,m
2];若存在,求出m,n的值;若不存在,说明理由.
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