Question

（本小題滿分14分）

         已知函數．

         （Ⅰ）當時，求函數的圖象在處的切線方程；

         （Ⅱ）判斷函數的單調性；

         （Ⅲ）若函數在上為增函數，求的取值范圍．

 

Answer 1

【答案】

（Ⅰ）．

（Ⅱ）當時，函數在單調遞增；

當時，函數在單調遞減，在上單調遞增．

（Ⅲ）．

【解析】(I)當a=2時，先求出的值，即切線的斜率，然后寫出點斜式方程，再化成一般式即可.

(II)先求導，可得,然后再對和a<0兩種情況進行討論研究其單調性.

(III)本小題轉化為在上恒成立，也可考慮求出f(x)的增區間D，然后根據求解也可.

（Ⅰ）當時，（），········································· 1分

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

∴ ，所以所求的切線的斜率為3．······················································· 3分

又∵，所以切點為.

 故所求的切線方程為：．······································································· 4分

（Ⅱ）∵，

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

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

②當時，

由，得；由，得；·························· 8分

綜上，當時，函數在單調遞增；

當時，函數在單調遞減，在上單調遞增．········ 9分

（Ⅲ）①當時，由（Ⅱ）可知，函數在單調遞增．此時，，故在上為增函數．······································································································· 11分

②當時，由（Ⅱ）可知，函數在上單調遞增．

∵　在上為增函數，

∴　，故，解得，

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

綜上所述，的取值范圍為．                      14分