已知函数f(x)=lnx-ax(a∈R).(1)求函数f(x)的单调区间;(2)当a>0时,求函数f(x)在[1,2]上的最小值
已知函数f(x)=lnx-ax(a∈R).(1)求函数f(x)的单调区间;(2)当a>0时,求函数f(x)在[1,2]上的最小值....
已知函数f(x)=lnx-ax(a∈R).(1)求函数f(x)的单调区间;(2)当a>0时,求函数f(x)在[1,2]上的最小值.
展开
擜誗劝
2015-01-06
·
超过49用户采纳过TA的回答
关注
(1)单调增区间是 ![](https://iknow-pic.cdn.bcebos.com/b3b7d0a20cf431adcce61e8a4836acaf2edd9848?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,单调减区间是 ![](https://iknow-pic.cdn.bcebos.com/34fae6cd7b899e51746468a741a7d933c9950d9e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) (2)当0<a<ln2时,最小值是-a;当a≥ln2时,最小值是ln2-2a. |
①知函数解析式求单调区间,实质是求f′(x)>0,f′(x)<0的解区间,并注意定义域; ②先研究f(x)在[1,2]上的单调性,再确定最值是端点值还是极值; ③由于解析式中含有参数a,要对参数a进行分类讨论. 规范(1)f′(x)= ![](https://iknow-pic.cdn.bcebos.com/b3119313b07eca8099306454922397dda1448348?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) -a(x>0).(1分) ①当a≤0时,f′(x)= ![](https://iknow-pic.cdn.bcebos.com/b3119313b07eca8099306454922397dda1448348?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) -a≥0,即函数f(x)的单调增区间是(0,+∞).(3分) ②当a>0时,令f′(x)= ![](https://iknow-pic.cdn.bcebos.com/b3119313b07eca8099306454922397dda1448348?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) -a=0,得x= ![](https://iknow-pic.cdn.bcebos.com/0e2442a7d933c8955194cac8d21373f08302009e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,当0<x< ![](https://iknow-pic.cdn.bcebos.com/0e2442a7d933c8955194cac8d21373f08302009e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) 时,f′(x)= ![](https://iknow-pic.cdn.bcebos.com/962bd40735fae6cd025ef9d20cb30f2443a70f9e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) >0,当x> ![](https://iknow-pic.cdn.bcebos.com/0e2442a7d933c8955194cac8d21373f08302009e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) 时,f′(x)= ![](https://iknow-pic.cdn.bcebos.com/962bd40735fae6cd025ef9d20cb30f2443a70f9e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) <0,所以函数f(x)的单调增区间是 ![](https://iknow-pic.cdn.bcebos.com/b3b7d0a20cf431adcce61e8a4836acaf2edd9848?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ,单调减区间是 ![](https://iknow-pic.cdn.bcebos.com/34fae6cd7b899e51746468a741a7d933c9950d9e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) .(6分) (2)①当 ![](https://iknow-pic.cdn.bcebos.com/0e2442a7d933c8955194cac8d21373f08302009e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ≤1,即a≥1时,函数f(x)在区间[1,2]上是减函数, 所以f(x)的最小值是f(2)=ln2-2a.(8分) ②当 ![](https://iknow-pic.cdn.bcebos.com/0e2442a7d933c8955194cac8d21373f08302009e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) ≥2,即0<a≤ ![](https://iknow-pic.cdn.bcebos.com/c995d143ad4bd113f027e58159afa40f4afb059e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) 时,函数f(x)在区间[1,2]上是增函数, 所以f(x)的最小值是f(1)=-a.(10分) ③当1< ![](https://iknow-pic.cdn.bcebos.com/0e2442a7d933c8955194cac8d21373f08302009e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) <2,即 ![](https://iknow-pic.cdn.bcebos.com/c995d143ad4bd113f027e58159afa40f4afb059e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) <a<1时,函数f(x)在区间 ![](https://iknow-pic.cdn.bcebos.com/d833c895d143ad4b53c4147381025aafa50f069e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) 上是增函数,在区间 ![](https://iknow-pic.cdn.bcebos.com/203fb80e7bec54e7c454a451ba389b504ec26ac1?x-bce-process=image/quality,q_85) 上是减函数, 又f(2)-f(1)=ln2-a, 所以当 ![](https://iknow-pic.cdn.bcebos.com/c995d143ad4bd113f027e58159afa40f4afb059e?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto) <a<ln2时,最小值是f(1)=-a; 当ln2≤a<1时,最小值是f(2)=ln2-2a.(12分) 综上可知,当0<a<ln2时,最小值是-a; 当a≥ln2时,最小值是ln2-2a.(14分) |
收起
为你推荐: