Question

已知函數(shù)．

(Ⅰ)討論函數(shù)在定義域內(nèi)的極值點(diǎn)的個(gè)數(shù)；

(Ⅱ)已知函數(shù)在處取得極值，且對,恒成立，

求實(shí)數(shù)的取值范圍．

 

Answer 1

【答案】

（Ⅰ）在上遞減，在上遞增，即在處有極小值．

∴當(dāng)時(shí)在上沒有極值點(diǎn)，當(dāng)時(shí)，在上有一個(gè)極值點(diǎn)．

（Ⅱ）．

【解析】（1）求出函數(shù)的導(dǎo)函數(shù)，在定義域內(nèi)研究其單調(diào)性就可得到極值點(diǎn)及其個(gè)數(shù)；（2）由函數(shù)在處取得極值，得，求出.把恒成立，轉(zhuǎn)化為分離參數(shù)求函數(shù)最值，即得的范圍.

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

當(dāng)時(shí)，在上恒成立，函數(shù)在單調(diào)遞減，

∴在上沒有極值點(diǎn)；················································································ 3分

當(dāng)時(shí)，得，得，

∴在上遞減，在上遞增，即在處有極小值．················ 5分

∴當(dāng)時(shí)在上沒有極值點(diǎn)，當(dāng)時(shí)，在上有一個(gè)極值點(diǎn)．

（Ⅱ）∵函數(shù)在處取得極值，∴，

∴，·········································································· 8分

令，可得在上遞減，在上遞增，··················· 10分

∴，即．       12分