【文档说明】2023高考数学科学复习创新方案（新高考题型版） 第4章 第4讲　第2课时　利用导数解决不等式恒（能）成立问题 含解析.doc，共(21)页，436.875 KB，由envi的店铺上传

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12()1(2021·)f(x)(x4)ex312x23x72g(x)aexcosxaR.(1)f(x)f(x)>0(2)a1x>0g(x)>2(3)max{mn}mnh(x)max{f(x)g(x)}h(x)≥0(0∞)a(1)f′(x)(x3)ex3x3(x3)·(

ex31)x>3x3>0ex31>0f′(x)>0x<3x3<0ex31<0f′(x)>0x3f′(x)0xRf′(x)≥0f(x)Rf(3)0f(x)>0(3∞)(2)a1g(x)excosxg′(x)exsin

x.x>0ex>1sinx[1,1]g′(x)exsinx>0g(x)(0∞)g(x)>g(0)2g(x)>2.(3)(1)x≥3f(x)≥0h(x)≥0x<3f(x)<0h(x)max{f(x)g(x)}h(x)≥0g(x)≥0(0,3)g(x)aexcosx≥0a≥cos

xex.r(x)cosxexx[0,3]2r′(x)sinxcosxex.r′(x)0x3π4.xr′(x)r(x)x03π43π43π43r′(x)0r(x)r(x)03π43π43r(x)(0,3)r

3π422e-3π4a≥22e-3π4.a22e-3π4∞.(1)f(x)≥g(x)“”h(x)f(x)g(x)“”u(x)g(x)f(x)h(x)min≥0u(x)max≤0(2)av(x)a≥v(x)xDa≥v(x)maxa≤v(x)xDa≤v(x)

min.1.(2021·)f(x)xlnxx≥1f(x)≥ax1a()f(x)≥ax1[1∞)a≤lnx1xx[1∞)a≤lnx1xminx[1∞)g(x)lnx1x(x≥1)3g′(x)1x1x2x1x2.g′(x)0x1.x

≥1g′(x)≥0g(x)[1∞)g(x)[1∞)g(1)1.a(∞1]()x1f(1)≥a1a1≤0a≤1.F(x)f(x)(ax1)xlnxax1F(x)≥0x≥1⇔F(x)min≥0x[1∞)F′(x)lnx1a≥0x[1∞)f(x)[1∞)F(x)minF(1)1a≥0a≤

1.a(∞1]2f(x)ax(2a1)lnx2xg(x)2alnx2xaR.(1)a>0f(x)(2)x1ee2f(x)≥g(x)a(1)f(x)(0∞)f′(x)a2a1x2x2ax2(2a1)x2x2(ax1)(x2)

x2.1a2a12x>0f′(x)≥0f(x)(0∞)0<1a<2a>124f′(x)>00<x<1ax>2f′(x)<01a<x<2.f(x)01a(2∞)1a2.1a>20<a<12f′(x)>00<x<2x>1af′(x)<02<x<1a.f(x)(0,2

)1a∞21a.(2)f(x)≥g(x)axlnx≥0a≥lnxxx1ee2.h(x)lnxxx1ee2a≥h(x)minh′(x)1lnxx2h′(x)0xe1ee2.1e

≤x<eh′(x)>0e<x≤e2h′(x)<0.h(x)1ee(ee2]h(x)x1exe2h1eeh(e2)2e2h1e<h(e)h(x)minh1eea≥e.a[e∞)5

12(1)x[ab]f(x)≥m⇔f(x)max≥m.(2)x[ab]f(x)≤m⇔f(x)min≤m.2.(2021·)f(x)xalnxg(x)a1x(aR)[1e]x0f(x0)<g(x0)a[f(x)g(x)]min<0x[1e]h(x)f(x)g(x)xalnx

a1xx[1e]h′(x)1axa1x2x2ax(a1)x2[x(a1)](x1)x2.a1≤1a≤0h′(x)≥0h(x)h(x)minh(1)a2<0a<21<a1<e0<a<e1h(x)[1a1)(a1e]h(x)minh(a1)(a1)aln(

a1)1a[1ln(a1)]2>2a1≥ea≥e1h′(x)≤0h(x)[1e]h(x)minh(e)eaa1e<0a>e21e1>e1a(∞2)e21e1∞.()63f(x)axxlnxg(x)x3x23.(1)x1x2[0,2]g(x1

)g(x2)≥MM(2)st122f(s)≥g(t)a(1)x1x2[0,2]g(x1)g(x2)≥Mg(x)maxg(x)min≥M.g(x)x3x23g′(x)3x22x3xx23.g′(x)0x0x23x[0,2]g(x)023232g

(x)ming238527g(x)maxg(2)1.g(x)maxg(x)min11227≥MM4.(2)st122f(s)≥g(t)122f(x)min≥g(x)max.(1)122g(x)g(2)

1.122f(x)axxlnx≥1a≥xx2lnxh(x)xx2lnxh′(x)12xlnxxφ(x)12xlnxxφ′(x)(2lnx3)x122φ′(x)<0h′(x)122h′(1)01x2h′(x)0712x1h′(x)0.h(x)xx2lnx

121(1,2]h(x)maxh(1)1a[1∞)(1)∀x1[ab]x2[cd]f(x1)>g(x2)⇔f(x)[ab]>g(x)[cd](2)∃x1[ab]x2[cd]f(x1)>g(x2)⇔f(x)[ab]>g(x)[cd](3)∀x1[ab]∃x2[cd]f(x1)>g(x2

)⇔f(x)[ab]>g(x)[cd](4)∃x1[ab]∀x2[cd]f(x1)>g(x2)⇔f(x)[ab]>g(x)[cd]3.f(x)(x1)ex1mx2g(x)x34xmx0<m≤6.x1Rx2(0,2]f(x1)≤g(x2)mf(x)(x1)ex1mx2f′(x)ex1(x1)ex1

2mxx(ex12m)0<m≤6ex1>0ex12m>0x>0f′(x)>0x<0f′(x)<0.f(x)(∞0)(0∞)f(x)minf(0)e.g′(x)3x24x2m≥43m0<m≤6g′(x)>0g(x)(0,2]g(x)maxg(2)822m62m.f(x)m

in≤g(x)max62m≥e80<m≤60<m≤3e2.m03e2.exlnx[]1(2021·)f(x)exalnx(aR)(1)aef(x)(2)a>0f(x)≥a(2lna)(1)aef(x)exelnx(x>0)f′(x)exexf′(x)

f′(1)0x(0,1)f′(x)<0x(1∞)f′(x)>0.f(x)(0,1)(1∞)(2)f′(x)exaxa>0f′(x)(0∞)a>ef′(1)ea<0aef′(1)ea0a<ef′aeeaee<0f′(a)ea1>0x0(0∞)f

′(x0)ex0ax00x0lnalnx0x(0x0)f′(x)<0g(x)x(x0∞)f′(x)>0g(x)f(x)≥f(x0)ex0alnx0ax0ax0alna≥2aalnaf(x)≥a(2lna)9(2021·)f(x)a(1x)exaRa≠0.(1)f(x)(

2)a1F(x)x213x3f(x)x0F(x)F(x0)232.(1)f(x)a(1x)exf′(x)axexa>0f′(x)>0x<0f′(x)<0x>0f(x)(∞0)(0∞)a<0f′(x)>0x>0f′

(x)<0x<0f(x)(∞0)(0∞)a>0f(x)(∞0)(0∞)a<0f(x)(∞0)(0∞)(2)a1F(x)x213x3(1x)exF′(x)x(2xex)h(x)2xexh′(x)1ex<0h(x)Rh(0)1>0h(

1)1e<0x0(0,1)h(x0)0ex02x0x(∞0)F′(x)<0F(x)x(0x0)F′(x)>0F(x)x(x0∞)F′(x)<0F(x)x0F(x)F(x0)x2013x30(1x0)ex

013x302x203x02x0(0,1)G(x)13x32x23x2x(0,1)G′(x)(x1)(x3)<0G(x)(0,1)G(0)2G(1)23F(x0)232.[ex≥x1lnx≤x1]2(2021·)mRf(x)e2xln(2x

m)(1)x0f(x)mf(x)10(2)m≥2f(x)>0.(1)f′(x)2e2x22xmx0f(x)f′(0)2e02m0m1.f(x)e2xln(2x1)x12∞f′(x)2e2x22x1f′(x)12∞f′(0)0f′(x)f(x)x1200(0∞)

f′(x)0f(x)m1f(x)120(0∞)(2)ex≥x1xRg(x)ex(x1)g′(x)ex1x(∞0)g′(x)<0g(x)x(0∞)g′(x)>0g(x)g(x)≥g(0)0e

x≥x1e2x≥2x1x0lnx≤x1ln(2xm)≥(2xm)12xm1e2xln(2xm)≥2x1(2xm)1m211x02xm1m1m≥2m≠1e2xln(2xm)>2m≥0(2xm>0)m1e2xln(2x1)≥2x1(2x1)1

1>0(2x1>0)m≥2f(x)>0(1)x≥1lnx(x>0)x1(2)ex≥x1(xR)x0ex>x1>x1>lnx(x>0x≠1)()f(x)exa.(1)f(x)lyx1a(2)f(x)lnx>0a(1)f′(x)exf(x)yx1f′(x)1ex1x0f(0)1a2.(2)e

x≥x1F(x)exx1F′(x)ex1F′(x)0x0x(∞0)F′(x)<0x(0∞)F′(x)>0F(x)(∞0)(0∞)F(x)minF(0)0F(x)≥0ex≥x1ex2≥x112x0lnx≤x1x1ex2>lnxa

≤2lnx<ex2≤exaa≤2f(x)lnx>0a≥3x1exa<lnxexa>lnxa2.1xx33x29x2≥mx[2,2]m()A(∞7]B(∞20]C(∞0]D[12,7]Bf(x)x33x29x2f′(x)3x26x9f′(x)0x13.f(1)7f(2)0f(2)2

0f(x)f(2)20m≤20.2f(x)xexg(x)(x1)2a∃x1x2Rf(x2)≤g(x1)a()A1e∞B[1∞)C[e∞)D1e∞Df′(x)exxex(1x)exx>1f′(x)>0f(x)x<1f′(x)<0f(x)x1f(x)13f(1)1e.

g(x)a.∃x1x2Rf(x2)≤g(x1)g(x)f(x)a≥1e.D.3(2019·)aRf(x)x22ax2ax≤1xalnxx>1.xf(x)≥0Ra()A[0,1]B[0,2]C[0e]D[1e]Cx≤1f(x)x22ax2a≥0f(x)xaa≥1f(x)minf(1)1>0

a<1f(x)minf(a)2aa2≥00≤a<1.a≥0.x>1f(x)xalnx≥0a≤xlnxg(x)xlnxg′(x)lnx1(lnx)2.g′(x)0xe1<x<eg′(x)<0x>eg′(x)

>0g(x)ming(e)ea≤e.a0≤a≤e[0e]C4(2021·)xexx3xalnx≥1x(1∞)a()A(∞1e]B(∞3]C(∞2]D(∞2e2]Ba≤exx3x1lnxx3exx1lnxe3lnxexx1lnxe

x3lnxx1lnx.f(x)exx1f′(x)ex1(∞0)[0∞)f(x)minf(0)0ex≥x1.ex3lnxx1lnx≥x3lnx1x1lnx3a≤3.B.145f(x)excos2x()Af(x)0π2Bf(x)0π2C∀x≥0f(x)≥0D∃x≥0f(x)

<0ACf(x)excos2xex12(1cos2x)f′(x)exsin2xx0π2sin2x>0ex>0f′(x)>0f(x)0π2ABx0π2f(x)≥f(0)0x≥π2f(x)excos2x>e1∀x≥0f(x)≥0CDAC6(2021·

)x>0f(x)x2exg(x)(m21)xlnxf(x)≥g(x)m________.[ee]f(x)≥g(x)⇔x2ex≥(m21)xlnx(x>0)⇔m21≤x2exlnxx(x>0)h(x)x2exlnxxxexxlnxxh′(x)(x1)exx2lnx1x2t

(x)(x1)exx2lnx1(x>0)t′(x)xex2x1x>0yt(x)(0∞)t(1)h′(1)0x(0,1)t(x)<0h′(x)<0h(x)(0,1)x(1∞)t(x)>0h′(x)>0h(x)(1∞)h(x)minh(1)1em21≤1ee≤m≤e.157

(2021·)lnx1x≤axbx(0∞)a0b________a>0ba________.11ea0b≥lnx1xmaxf(x)lnx1xf′(x)1lnx1x2lnxx2f′(x)0x10<x<1f′(x

)>0f(x)x>1f′(x)<0f(x)f(x)maxf(1)ln1111b≥1b1.f(x)lnx1xa>0axb0xbabayaxbxlnx1x≤axb.f(x)lnx1x0x1eyaxbxba≤1eba≥

1ebamin1e.168(2021·)f(x)2(x1)ex.(1)f(x)(a∞)f(a)(2)g(x)exxpx0[1e]g(x0)≥f(x0)x0p(1)f′(x)2xex>0x>0f(x

)(0∞)a≥0f(a)≥f(0)2.f(a)[2∞)(2)x0[1e]g(x0)≥2(x01)ex0x0x0[1e]p≥(2x03)ex0h(x)(2x3)exp≥h(x)minh′(x)(2x1)ex.x≥12x1≥1ex>0h′(x)>0h(x)(2x

3)ex[1e]h(x)minh(1)ep≥ep[e∞)9af(x)alnxx24x.(1)3f(x)a(2)g(x)(a2)xx01eef(x0)≤g(x0)a(1)f(x)(0∞)f′(x

)ax2x42x24xax.3f(x)f′(3)0a6.a63f(x)a6.(2)f(x0)≤g(x0)(x0lnx0)a≥x202x0F(x)xlnx(x>0)F′(x)x1x(x>0)170<x<1F′(x)<0F(x)x>1F′(x)>0F(x)F(x)>F(1)1>0a≥x202x0

x0lnx0.G(x)x22xxlnxx1eeG′(x)(2x2)(xlnx)(x2)(x1)(xlnx)2(x1)(x2lnx2)(xlnx)2.x1ee22lnx2(1lnx)≥0x2lnx2>0x1e1G′(x)<0G(x)x(1e)G′(x)>0G(

x)G(x)minG(1)1a≥G(x)min1a[1∞)10(2021·)f(x)exmx2.(1)xyf(x)m(2)x≥0f(x)≥2xsinx1m(1)yf(x)(x0,0)f′(x)ex2mxf′(x0)ex02mx00ex02

mx00f(x0)0ex0mx200me24.(2)g(x)f(x)(2xsinx1)exmx22xsinx1x≥0g′(x)ex2mx2cosx18h(x)g′(x)h′(x)ex2msinxt(x)h′(x)t′(x)excosx≥0h′(x)[0∞)h′(x)≥h′(0)12m.m≤12

h′(x)≥h′(0)12m≥0g′(x)[0∞)g′(x)≥g′(0)0g(x)[0∞)g(x)≥g(0)0m≤12m>12h′(0)12m<0h′(2m)e2m2msin2m>1sin2m≥0h′(x)[0∞)x0h′(x0)0.x(0x0)x0(x0∞)h

′(x)0h(x)g′(x)x[0x0]g′(x)≤g′(0)0g(x)[0x0]g(x)≤g(0)0m∞12.11(2021·)f(x)ex12x2ax1g(x)cosx12x21.(1)a1x≥0f(x)

≥0(2)f(x)g(x)≥0[0∞)a(1)a1f(x)ex12x2x1f′(x)exx1φ(x)f′(x)φ′(x)ex1≥0f′(x)[0∞)f′(x)≥f′(0)0f(x)[0∞)19f(x)≥f(0)0(2)f(x)g(x)exc

osxax2h(x)f(x)g(x)excosxax2h(x)≥0h′(x)exsinxat(x)h′(x)t′(x)excosxex≥11≤cosx≤1t′(x)≥0h′(x)[0∞)h′(x)≥h′(0)1a1

a≥0a≤1h′(x)≥0h(x)[0∞)h(x)≥h(0)0.a≤11a<0a>1h′(0)1a<0x→∞h′(x)→∞∃x0(0∞)h′(x0)0x(0x0)h′(x)<0h(x)(0x0)h(x)<h(0)0a(∞1]12(2021·)f(x)12lnxmxg(x)xax(a>0)(1)f(

x)(2)m12e2∀x1x2[2,2e2]g(x1)≥f(x2)a(1)f(x)12lnxmxx>0f′(x)12xmm≤0f′(x)>0f(x)(0∞)m>0f′(x)0x12mf′(x)>0x>00<x<12mf′(x)<0x>0x>12m.20f(x

)012m12m∞m≤0f(x)(0∞)m>0f(x)012m12m∞.(2)m12e2f(x)12lnx12e2x.∀x1x2[2,2e2]g(x1)≥f(x2)x[2,2e2]g(x)min≥f(

x)max(1)[2,2e2]f(x)f(e2)12g′(x)1ax2>0(a>0)x[2,2e2]g(x)[2,2e2]g(x)ming(2)2a2.2a2≥12a≤3a>0a(0,3]a(0,3]13(2021·)f(x)axlnx

bx2ax.(1)yf(x)(1f(1))xy120ab(2)a≤0b12∀x1x2(1e)|f(x1)f(x2)||x1x2|<3a(1)f′(x)a(1lnx)2bxaalnx2bxf′(1)2b1b12f(1)ba32a1.a1

b12.(2)a≤0b12f′(x)alnxx<0f(x)(1e)x1<x2f(x1)>f(x2)f(x1)f(x2)x2x1<3.21f(x1)f(x2)<3x23x1f(x1)3x1<f(x2)3x

2.g(x)f(x)3xg(x)(1e)g′(x)f′(x)3alnxx3≥0(1e)a≥x3lnxx(1e)h(x)x3lnxx(1e)h′(x)lnx3x1(lnx)2t(x)lnx3x1t′(x

)1x3x2x3x2<0.t(x)(1e)t(x)>t(e)3eh′(x)>0h(x)(1e)h(x)<h(e)e3a≥e3.e3≤a≤0.