Question

設(shè)函數(shù)在，處取得極值，且．
（Ⅰ）若，求的值，并求的單調(diào)區(qū)間；
（Ⅱ）若，求的取值范圍．

Answer 1

解：．①····················································· 2分
（Ⅰ）當(dāng)時(shí)，；
由題意知為方程的兩根，所以．
由，得．········································································· 4分
從而，．
當(dāng)時(shí)，；當(dāng)時(shí)，．
故在單調(diào)遞減，在，單調(diào)遞增．····························· 6分
（Ⅱ）由①式及題意知為方程的兩根，
所以．從而，
由上式及題設(shè)知．······································································· 8分
考慮，．………………………10分
故在單調(diào)遞增，在單調(diào)遞減，從而在的極大值為．
又在上只有一個(gè)極值，所以為在上的最大值，且最小值為．所以，即的取值范圍

略