專(zhuān)題22隱零點(diǎn)問(wèn)題在求解函數(shù)問(wèn)題時(shí)，很多時(shí)候都需要求函數(shù)f(x)在區(qū)間I上的零點(diǎn)，但所述情形都難以求出其準(zhǔn)確值，導(dǎo)致解題過(guò)程將無(wú)法繼續(xù)進(jìn)行．但可這樣嘗試求解：先證明函數(shù)f(x)在區(qū)間I上存在唯一的零點(diǎn)(例如，函數(shù)f(x)在區(qū)間I上是單調(diào)函數(shù)且在區(qū)間I的兩個(gè)端點(diǎn)的函數(shù)值異號(hào)時(shí)就可證明存在唯一的零點(diǎn))，這時(shí)可設(shè)出其零點(diǎn)是x0.因?yàn)閤0不易求出(當(dāng)然，有時(shí)是可以求出但無(wú)需求出)，所以把零點(diǎn)x0叫做隱零點(diǎn)；若x0容易求出，就叫做顯零點(diǎn)，而后解答就可繼續(xù)進(jìn)行．實(shí)際上，此解法類(lèi)似于解析幾何中“設(shè)而不求”的方法．1．設(shè)函數(shù)f(x)＝ex－ax－2.(1)求f(x)的單調(diào)區(qū)間；(2)若a＝1，k為整數(shù)，且當(dāng)x>0時(shí)，(x－k)f′(x)＋x＋1>0，求k的最大值．2．已知函數(shù)f(x)＝eq\f(1－lnx,x2).(1)求函數(shù)f(x)的零點(diǎn)及單調(diào)區(qū)間；(2)求證：曲線y＝eq\f(lnx,x)存在斜率為6的切線，且切點(diǎn)的縱坐標(biāo)y0<－1.3．設(shè)函數(shù)f(x)＝e2x－alnx.(1)討論f(x)的導(dǎo)函數(shù)f′(x)零點(diǎn)的個(gè)數(shù)；(2)求證：當(dāng)a＞0時(shí)，f(x)≥2a＋alneq\f(2,a).4．已知函數(shù)f(x)＝xex－a(x＋lnx)．(1)討論f(x)極值點(diǎn)的個(gè)數(shù)；(2)若x0是f(x)的一個(gè)極小值點(diǎn)，且f(x0)>0，證明：f(x0)>2(x0－xeq\o\al(3,0))．5．已知函數(shù)f(x)＝－lnx－x2＋x，g(x)＝(x－2)ex－x2＋m(其中e為自然對(duì)數(shù)的底數(shù))．當(dāng)x∈(0,1]時(shí)，f(x)>g(x)恒成立，求正整數(shù)m的最大值．6．已知f(x)＝x2－4x－6lnx.(1)求f(x)在(1，f(1))處的切線方程以及f(x)的單調(diào)性；(2)對(duì)任意x∈(1，＋∞)，有xf′(x)－f(x)＞x2＋6k·eq\b\lc\(\rc\)(\a\vs4\al\co1(1－\f(1,x)))－12恒成立，求k的最大整數(shù)解；(3)令g(x)＝f(x)＋4x－(a－6)lnx，若g(x)有兩個(gè)零點(diǎn)分別為x1，x2(x1＜x2)且x0為g(x)的唯一的極值點(diǎn)，求證：x1＋3x2＞4x0.7．已知函數(shù)SKIPIF1<0．當(dāng)SKIPIF1<0時(shí),SKIPIF1<0,求整數(shù)SKIPIF1<0的最大值．8．已知函數(shù)SKIPIF1<0.（1）證明：函數(shù)SKIPIF1<0在SKIPIF1<0上存在唯一的零點(diǎn)；（2）若函數(shù)SKIPIF1<0在區(qū)間SKIPIF1<0上的最小值為1，求SKIPIF1<0的值.9．已知函數(shù)SKIPIF1<0，SKIPIF1<0．（1）令SKIPIF1<0，求SKIPIF1<0的最小值；（2）若SKIPIF1<0恒成立，求SKIPIF1<0的取值范圍．10．已知函數(shù)SKIPIF1<0的圖象在點(diǎn)SKIPIF1<0（SKIPIF1<0為自然對(duì)數(shù)的底數(shù)）處的切線斜率為SKIPIF1<0．（1）求實(shí)數(shù)SKIPIF1<0的值；（2）若SKIPIF1<0，且SKIPIF1<0對(duì)任意SKIPIF1<0恒成立，求SKIPIF1<0的最大值．11．已知函數(shù)SKIPIF1<0.（1）若SKIPIF1<0在SKIPIF1<0處的切線斜率為SKIPIF1<0,求實(shí)數(shù)a的值；（2）當(dāng)SKIPIF1<0時(shí),判斷SKIPIF1<0的極值點(diǎn)個(gè)數(shù)；（3）對(duì)任意SKIPIF1<0,有SKIPIF1<0,求a的取值范圍.12．已知定義在SKIPIF1<0上的函數(shù)SKIPIF1<0.（1）求證：SKIPIF1<0存在唯一的零點(diǎn),且零點(diǎn)屬于SKIPIF1<0；（2）若k∈Z,且SKIPIF1<0對(duì)任意的SKIPIF1<0恒成立,求k的最大值.13．已知函數(shù)SKIPIF1<0,SKIPIF1<0．（1）設(shè)函數(shù)SKIPIF1<0,求SKIPIF1<0的最大值；（2）證明：SKIPIF1<0．14．已知函數(shù)SKIPIF1<0,在定義域上有兩個(gè)極值點(diǎn)SKIPIF1<0．（1）求實(shí)數(shù)a的取值范圍；（2）求證:SKIPIF1<015．已知函數(shù)SKIPIF1<0．（SKIPIF1<0）（1）討論SKIPIF1<0的單調(diào)性；（2）若對(duì)任意SKIPIF1<0都有SKIPIF1<0,求實(shí)數(shù)a的取值范圍．16．已知函數(shù)SKIPIF1<0（1）討論SKIPIF1<0的單調(diào)性；（2）若關(guān)于SKIPIF1<0的不等式SKIPIF1<0在SKIPIF1<0上恒成立,求實(shí)數(shù)SKIPIF1<0的取值范圍.17．已知函數(shù)SKIPIF1<0,SKIPIF1<0為SKIPIF1<0的導(dǎo)函數(shù)．（1）討論SKIPIF1<0在區(qū)間SKIPIF1<0內(nèi)極值點(diǎn)的個(gè)數(shù)；（2）若SKIPIF1<0,SKIPIF1<0時(shí),SKIPIF1<0恒成立,求整數(shù)SKIPIF1<0的最小值．18．已知函數(shù)SKIPIF1<0,SKIPIF1<0.（1）若函數(shù)SKIPIF1