Question

（本小题满分14分）

         已知函数．

         （Ⅰ）当时，求函数的图象在处的切线方程；

         （Ⅱ）判断函数的单调性；

         （Ⅲ）若函数在上为增函数，求的取值范围．

Answer

（Ⅰ）．

（Ⅱ）当时，函数在单调递增；

当时，函数在单调递减，在上单调递增．

（Ⅲ）．

【解析】(I)当a=2时，先求出的值，即切线的斜率，然后写出点斜式方程，再化成一般式即可.

(II)先求导，可得,然后再对和a<0两种情况进行讨论研究其单调性.

(III)本小题转化为在上恒成立，也可考虑求出f(x)的增区间D，然后根据求解也可.

（Ⅰ）当时，（），········································· 1分

∴，···································································· 2分

∴ ，所以所求的切线的斜率为3．······················································· 3分

又∵，所以切点为.

 故所求的切线方程为：．······································································· 4分

（Ⅱ）∵，

∴······························································· 5分

①当时，∵，∴；····························································· 6分

②当时，

由，得；由，得；·························· 8分

综上，当时，函数在单调递增；

当时，函数在单调递减，在上单调递增．········ 9分

（Ⅲ）①当时，由（Ⅱ）可知，函数在单调递增．此时，，故在上为增函数．······································································································· 11分

②当时，由（Ⅱ）可知，函数在上单调递增．

∵ 在上为增函数，

∴ ，故，解得，

∴ ．······························································································ 13分

综上所述，的取值范围为．                      14分