题目内容

设函数处取得极值,且
(Ⅰ)若,求的值,并求的单调区间;
(Ⅱ)若,求的取值范围.
解:.①····················································· 2分
(Ⅰ)当时,
由题意知为方程的两根,所以
,得.········································································· 4分
从而
时,;当时,
单调递减,在单调递增.····························· 6分
(Ⅱ)由①式及题意知为方程的两根,
所以.从而
由上式及题设知.······································································· 8分
考虑.………………………10分
单调递增,在单调递减,从而的极大值为
上只有一个极值,所以上的最大值,且最小值为.所以,即的取值范围
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