Question

已知函數(shù)(為常數(shù),是自然對(duì)數(shù)的底數(shù)),曲線在點(diǎn)處的切線與軸平行．

(Ⅰ)求的值;

(Ⅱ)求的單調(diào)區(qū)間;

(Ⅲ)設(shè),其中為的導(dǎo)函數(shù)．證明:對(duì)任意．

 

Answer 1

(Ⅰ)1

(Ⅱ)在區(qū)間內(nèi)為增函數(shù);在內(nèi)為減函數(shù)．

(Ⅲ)見解析．

【解析】(Ⅰ)由f(x)=可得,而,即,解得k=1．

(Ⅱ),令可得x=1,

當(dāng)時(shí),;當(dāng)時(shí),．于是在區(qū)間內(nèi)為增函數(shù);在內(nèi)為減函數(shù)．(Ⅲ),(1)當(dāng)時(shí), ,．(2)當(dāng)時(shí),要證．只需證即可設(shè)函數(shù)

則,則當(dāng)時(shí)．

令解得,當(dāng)時(shí);當(dāng)時(shí),則當(dāng)時(shí),且,則,于是可知當(dāng)時(shí)成立綜合(1)(2)可知對(duì)任意x＞0,恒成立．另證1:設(shè)函數(shù),則,則當(dāng)時(shí),于是當(dāng)時(shí),要證,只需證即可,設(shè),,令解得,當(dāng)時(shí);當(dāng)時(shí),則當(dāng)時(shí),于是可知當(dāng)時(shí)成立綜合(1)(2)可知對(duì)任意x＞0,恒成立．另證2:根據(jù)重要不等式當(dāng)時(shí),即,于是不等式,設(shè),,令解得,當(dāng)時(shí);當(dāng)時(shí),則當(dāng)時(shí),于是可知當(dāng)時(shí)，成立．