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1、專題檢測(cè)(二十三) “函數(shù)與導(dǎo)數(shù)”壓軸大題的搶分策略1(2018武漢調(diào)研)已知函數(shù)f(x)ln x(aR)(1)討論函數(shù)f(x)的單調(diào)性;(2)當(dāng)a0時(shí),證明:f(x).解:(1)f(x)(x0)當(dāng)a0時(shí),f(x)0,f(x)在(0,)上單調(diào)遞增當(dāng)a0時(shí),若xa,則f(x)0,函數(shù)f(x)在(a,)上單調(diào)遞增;若0xa,則f(x)0時(shí),f(x)minf(a)ln a1.要證f(x),只需證ln a1,即證ln a10.令函數(shù)g(a)ln a1,則g(a)(a0),當(dāng)0a1時(shí),g(a)1時(shí),g(a)0,所以g(a)在(0,1)上單調(diào)遞減,在(1,)上單調(diào)遞增,所以g(a)ming(1)0.所以l
2、n a10恒成立,所以f(x).2(2018全國(guó)卷)已知函數(shù)f(x)exax2.(1)若a1,證明:當(dāng)x0時(shí),f(x)1;(2)若f(x)在(0,)只有一個(gè)零點(diǎn),求a.解:(1)證明:當(dāng)a1時(shí),f(x)1等價(jià)于(x21)ex10.設(shè)函數(shù)g(x)(x21)ex1,則g(x)(x22x1)ex(x1)2ex.當(dāng)x1時(shí),g(x)0,h(x)沒(méi)有零點(diǎn);()當(dāng)a0時(shí),h(x)ax(x2)ex.當(dāng)x(0,2)時(shí),h(x)0.所以h(x)在(0,2)上單調(diào)遞減,在(2,)上單調(diào)遞增故h(2)1是h(x)在(0,)上的最小值當(dāng)h(2)0,即a時(shí),h(x)在(0,)上沒(méi)有零點(diǎn)當(dāng)h(2)0,即a時(shí),h(x)在(0
3、,)上只有一個(gè)零點(diǎn)當(dāng)h(2)時(shí),因?yàn)閔(0)1,所以h(x)在(0,2)上有一個(gè)零點(diǎn)由(1)知,當(dāng)x0時(shí),exx2,所以h(4a)11110,故h(x)在(2,4a)上有一個(gè)零點(diǎn)因此h(x)在(0,)上有兩個(gè)零點(diǎn)綜上,當(dāng)f(x)在(0,)上只有一個(gè)零點(diǎn)時(shí),a.3(2018西安質(zhì)檢)設(shè)函數(shù)f(x)ln x(kR)(1)若曲線yf(x)在點(diǎn)(e,f(e)處的切線與直線x20垂直,求f(x)的單調(diào)性和極小值(其中e為自然對(duì)數(shù)的底數(shù));(2)若對(duì)任意的x1x20,f(x1)f(x2)0),曲線yf(x)在點(diǎn)(e,f(e)處的切線與直線x20垂直,f(e)0,即0,得ke,f(x)(x0)由f(x)0,
4、得0x0,得xe,f(x)在(0,e)上單調(diào)遞減,在(e,)上單調(diào)遞增,當(dāng)xe時(shí),f(x)取得極小值,且f(e)ln e2.f(x)的極小值為2.(2)由題意知對(duì)任意的x1x20,f(x1)x10),則h(x)在(0,)上單調(diào)遞減,h(x)10在(0,)上恒成立,即當(dāng)x0時(shí),kx2x2恒成立,k.故k的取值范圍是.4(2018全國(guó)卷)已知函數(shù)f(x)(2xax2)ln(1x)2x.(1)若a0,證明:當(dāng)1x0時(shí),f(x)0時(shí),f(x)0;(2)若x0是f(x)的極大值點(diǎn),求a.解:(1)證明:當(dāng)a0時(shí),f(x)(2x)ln(1x)2x,f(x)ln(1x).設(shè)函數(shù)g(x)ln(1x),則g(x
5、).當(dāng)1x0時(shí),g(x)0時(shí),g(x)0,故當(dāng)x1時(shí),g(x)g(0)0,且僅當(dāng)x0時(shí),g(x)0,從而f(x)0,且僅當(dāng)x0時(shí),f(x)0.所以f(x)在(1,)上單調(diào)遞增又f(0)0,故當(dāng)1x0時(shí),f(x)0時(shí),f(x)0.(2)若a0,由(1)知,當(dāng)x0時(shí),f(x)(2x)ln(1x)2x0f(0),這與x0是f(x)的極大值點(diǎn)矛盾若a0,設(shè)函數(shù)h(x)ln(1x).由于當(dāng)|x|0,故h(x)與f(x)符號(hào)相同又h(0)f(0)0,故x0是f(x)的極大值點(diǎn),當(dāng)且僅當(dāng)x0是h(x)的極大值點(diǎn)h(x).若6a10,則當(dāng)0x,且|x|0,故x0不是h(x)的極大值點(diǎn)若6a10,則a2x24ax6a10存在根x10,故當(dāng)x(x1,0),且|x|min時(shí),h(x)0;當(dāng)x(0,1)時(shí),h(x)0.所以x0是 h(x)的極大值點(diǎn),從而x0是 f(x)的極大值點(diǎn)綜上,a