重难点04利用导数研究不等式恒（能）成立问题七大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点04 利用导数研究不等式恒（能）成立问题【七大题型】

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1
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2
𝑥−𝑎−1
e
1
𝑥2 +
e ≤0:¬
!MA¬
𝑎;Ā‹
12d'    )
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2e
Bx(−∞,0)
Cx −1
2e , + ∞
Dx[0, + ∞)
C% 1-2'2024·āĂ*ă·÷üýþ)Ÿ·𝑥;#$%(e−1)(ln𝑎+ 𝑥) ≥𝑎e𝑥−1•𝑥∈[0,1]¤:!MA¬

𝑎;Ā‹12J'    )
Ax 1
2e ,e2
Bx 1
e ,e
Cx 1
e ,e2
Dx 1
2e ,e
C% 1-3'2024·ĄąĆć·Ý÷)ÿt8
𝑓(𝑥) =
1
e𝑥+1−𝑥3M“·ĈU;𝑥∈(1,2]M#$%𝑓𝑥+1
𝑥−1 +𝑓
𝑡+1
(𝑥−1)2(𝑥−6) < 1&*+MA¬
𝑡;Ā‹12d'    )
Ax(1, + ∞)
Bx[−1,1]
Cx(−∞,−1]
Dx(−∞,−1)
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12
ð 2'2024·ĉĊċĂ·÷üýþ)ÿt𝑎∈𝑁∗M8
𝑓(𝑥) = e3𝑥−𝑥𝑎> 0&*+MA𝑎;Šm‹d'    )
Ax2
Bx3
Cx6
Dx7
      
      
C% 2-1'2024·āĂČč·ö÷)ÿt#$%𝑎𝑥e𝑥+𝑥> 1−ln𝑥:!MA¬
𝑎;Ā‹12d'ĎĎ)
Ax −1
e2 , + ∞
Bx −1
e , + ∞
Cx −∞, 1
e2
Dx −∞, 1
e
C% 2-2'2024·āĂ*ă·Ý÷)Ÿ𝑥∈[0, + ∞)M𝑥2 +𝑎𝑥+ 1 ≤e𝑥&*+MA¬
𝑎;Šm‹d'    )
Ax𝑒
Bx2
Cx𝑒−1
Dx𝑒−2
C% 2-3'2024·āĂďĐ·Ý÷)ÿt8
𝑓(𝑥) =
1
3𝑥3M𝑔(𝑥) = e𝑥−
1
2𝑥2−𝑥M∃𝑥1M𝑥2 ∈[1,2]Â
|𝑔(𝑥1)−𝑔(𝑥2)| > 𝑘|𝑓(𝑥1)−𝑓(𝑥2)|'𝑘dK
)*+MAK
𝑘;Ā‹12d'    )
Ax(−∞,e−2]
Bx(−∞,e−2)
Cx −∞, e2−3
4
Dx −∞, e2−3
4
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ð 3'2024·đĒēĔ·Ý÷)ÿt8
𝑓(𝑥) = ln𝑥−𝑎𝑥,𝑔(𝑥) =
2
𝑎𝑥,𝑎≠0x
(1)08
𝑓(𝑥);ĕĖ¹ºŽ
(2)Ÿ𝑓(𝑥) ≤𝑔(𝑥)&*+M0𝑎;Šė‹x
C% 3-1'2024·āĂĘć·ö÷)ÿt8
𝑓(𝑥) = 2𝑥3−𝑎𝑥2 +2(𝑎> 0)x
(1)0ęĚ𝑦= 𝑓(𝑥)•L(0,𝑓(0))¿;ěĚWXŽ
(2)Ÿ∃𝑥∈[−1,1]M|𝑓(𝑥)| ≥3M0¬
 a ;Ā‹12x
C% 3-2'2024·Ĝĝ·Ý÷)ÿt8
𝑓(𝑥) = ln(𝑥+ 1) +𝑘(𝑥+ 1)x
(1)0𝑓(𝑥);ĕĖ¹ºŽ
(2)Ÿ𝑓(𝑥) ≤−1&*+M0¬
𝑘;Ā‹12Ž
(3)0<
𝑛
𝑖=2
ln𝑖
𝑖+1 <
𝑛(𝑛−1)
4
x'𝑛∈𝑁Ê𝑛≥2)
      
      
C% 3-3'2024·āĂ·÷üýþ)ÿt8
𝑓(𝑥) = 𝑎𝑥ln𝑥−2𝑥+ 𝑏'𝑎M𝑏∈𝑅)•L(1,𝑓(1))¿;ěĚWX
d𝑦= −𝑥x
(1)08
𝑓(𝑥);ԋŽ
(2)Ğ𝑔(𝑥) = e𝑥𝑥𝑓1
𝑥+ 2 +𝑚𝑥'𝑚∈𝑅)MŸ𝑔(𝑥) ≥0&*+M0𝑚;Ā‹12x
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2𝑎
e𝑥0 + e2𝑥0−2*+MA𝑎;
Ā‹12J'    )
Ax 1
2e ,e2
Bx 1
e2 ,e2
Cx 1
e2 ,e4
Dx 1
e ,e4
C% 4-1'2024·ĄąĆć·ö÷)Ÿ· x ;#$%e𝑥+𝑥+ 2ln
1
𝑥≥𝑚𝑥2 + ln𝑚&*+MA¬
 m ;Šm
‹d'     )
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1
2
Bxe2
4
Cxe2
2
Dxe2
C% 4-2'2023·ĢďģĤ·÷üýþ)Ÿġ•𝑥∈[1, + ∞)M‚·𝑥;#$% 1 + 1
𝑥
𝑥+𝑎
≥e*+MA
¬
𝑎;Šė‹d'    )
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Bx
1
ln2
Cxln2−1
Dx
1
ln2−1
C% 4-3'2024·ĥùĦć·ö÷)ÿt8
𝑓(𝑥) = e𝑘𝑥+1M𝑔(𝑥) = 1 + 1
𝑥ln𝑥.Ÿ𝑘𝑓(𝑥) ≥𝑔(𝑥)MA k ;Ā
‹12d'    )
Ax(0,e]
Bx[e,+∞)
Cx 1
e , + ∞
Dx 0, 1
e
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ð 5'2024·ħďĨ=·Ý÷)ÿt8
𝑓(𝑥) = e𝑥−sin𝑎𝑥Ž
      
      
(1)Ü𝑎= −1nM<=
“ĈU𝑥∈−π
6 , + ∞M𝑓(𝑥) > 0Ž
(2)Ÿ𝑥= 0J8
𝑓(𝑥);ԋLM0¬
𝑎;‹x
C% 5-1'2024·ĩĪ·÷üýþ)ÿt8
𝑓(𝑥) = e𝑎𝑥+
1
2𝑥2−𝑎𝑥'𝑎∈R).
(1)Ü𝑎= 1nM0𝑓(𝑥);Š‹Ž
(2)Ü𝑎∈[−1,1]nM<=
“ĈU;𝑥1M𝑥2 ∈[−2,2]Mă:|𝑓(𝑥1)−𝑓(𝑥2)| ≤e2−1.
C% 5-2'2024·øùīĬ·÷üýþ)ÿt8
𝑓(𝑥) = 𝑎e𝑥(𝑎≠0),𝑔(𝑥) = 𝑥2,𝑔′(𝑥)d𝑔(𝑥);G8
.
(1)<=
Ü𝑎= 1nM∀𝑥∈(0, + ∞),𝑓(𝑥) > 𝑔′(𝑥)Ž
(2)Ÿ𝑓(𝑥)9𝑔(𝑥):z²ěĚM0 a ;Ā‹12.
C% 5-3'2024·ĭćĮįİ·Ý÷)Ÿ8
𝑓(𝑥)•[𝑎,𝑏]˜:©VMʓ·ĈU#ı;𝑥1,𝑥2 ∈[𝑎,𝑏]Mă:
|𝑓(𝑥1)−𝑓(𝑥2)| < 𝑘|𝑥1−𝑥2|MAá𝑓(𝑥)d[𝑎,𝑏]˜;“k 38
”
(1)Ÿ𝑓(𝑥) = 𝑥2Mš›𝑓(𝑥)Jœd[1,2]˜;“4 38
”Ž
(2)Ÿ𝑓(𝑥) =
2
eln𝑥+ (𝑎+ 1)𝑥+
1
𝑥d[1,e]˜;“2 38
”M0¬
 a ;Ā‹12Ž
(3)Ÿ𝑓(𝑥)d[1,2]˜;“2 38
”Ê𝑓(1) = 𝑓(2)M<=
∀𝑥1M𝑥2 ∈[1,2]M|𝑓(𝑥1)−𝑓(𝑥2)| < 1.
      
      
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ð 6'23-24 öÄ·IJij·Ĵĵ)Ÿ#$%sin𝑥> 𝑥−𝑎𝑥3“·𝑥∈(0,
𝜋
2)&*+M0𝑎;Ā‹12.
C% 6-1'2023 Ý·IJij·Ķķ)ÿt8
𝑓(𝑥)=𝑎ln𝑥+ 𝑏𝑥(𝑎,𝑏∈𝑅)•𝑥=
1
2¿Ā‚Ô‹MÊęĚ
𝑦= 𝑓(𝑥)•L(1,𝑓(1))¿;ěĚ9Ě𝑥−𝑦+ 1 = 0ĸ.
(1)0¬
𝑎,𝑏;‹Ž
(2)Ÿ∀𝑥∈[1 ,  +∞)M#$%𝑓(𝑥) ≤(𝑚−2)𝑥−
𝑚
𝑥&*+M0¬
𝑚;Ā‹12.
C% 6-2'2024·Ĺĥ·ö÷)①•ĺ-PM0ÔÕ:a{»o;
Ļļ——>?@ AM AP:
Ľ5
Ÿ8
𝑓(𝑥)M𝑔(𝑥);G8
-ľd𝑓′(𝑥)M𝑔′(𝑥)MÊlim
𝑥→𝑎𝑓(𝑥) = lim
𝑥→𝑎𝑔(𝑥) = 0MA
lim
𝑥→𝑎
𝑓(𝑥)
𝑔(𝑥) = lim
𝑥→𝑎
𝑓′(𝑥)
𝑔′(𝑥).
②Ğ𝑎> 0Mk Jm· 1 ;ªĿ
MŸ8
𝑓(𝑥)ž
“ĈU𝑥∈[0,𝑎]Mŀ:𝑓(𝑥) ≥𝑓𝑥
𝑘*+MÊlim
𝑥→0𝑓(𝑥)
= 0MAá8
𝑓(𝑥)d¹º[0,𝑎]˜; k ŁłŃńŅ8
.
Ľ]À˜z„ņMŇňÄÅ,
(1)iš›𝑓(𝑥) = 𝑥3−3𝑥Jœd¹º[0,3]˜; 2 ŁłŃńŅ8
Ž
(2)ʼnŊ
lim
𝑥→0(1 + 𝑥)
1
𝑥Ž
(3)<=
 sin𝑥
𝑥−π
3
< cos𝑥M𝑥∈π, 3
2 π .
      
      
C% 6-3'23-24 öÄ·đĒŋĪ·Ĵĵ)•Ōō8
,nMŎŏQKՃ08
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,Mő8
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ř A‡Šd
“Ğ8
𝑓(𝑥)M𝑔(𝑥)žÄŲ³
①lim
𝑥→𝑎𝑓(𝑥) = 0Mlim
𝑥→𝑎𝑔(𝑥) = 0Ž
②•L a ¿8
𝑓(𝑥)Ã𝑔(𝑥);šŢJéêÊţŤ;M¥8
𝑓(𝑥)Ã𝑔(𝑥)•L a ¿ġ•G
Ž
③lim
𝑥→𝑎
𝑓′(𝑥)
𝑔′(𝑥) = 𝐴MOP A J¸ť©¬
Ž
Alim
𝑥→𝑎
𝑓(𝑥)
𝑔(𝑥) = lim
𝑥→𝑎
𝑓′(𝑥)
𝑔′(𝑥) = 𝐴.”
ÌÍMŦĞ:8
𝑓(𝑥) = e𝑥M𝑔(𝑥) = 𝑡𝑥+ 1.
(1)Ÿ𝑓(𝑥) ≥𝑔(𝑥)&*+M0 t ;Ā‹12Ž
(2)<=
e𝑥−ln𝑥≥2.
 7  BCD;&'()*+,
ð 7'2023·āĂĘć·a÷)ÿt8
𝑓(𝑥) = 𝑎𝑥+ 1−𝑥ln𝑥;šŢ•𝑥= 1¿;ěĚ9Ě𝑥−𝑦= 0ŧ”x
(1)08
𝑓(𝑥);ĕĖ¹ºŽ
(2)Ÿ∀𝑥1,𝑥2 ∈(0, + ∞)MÊ𝑥1 > 𝑥2nM𝑓(𝑥1)−𝑓(𝑥2) > 𝑚𝑥2
1−𝑥2
2 M0¬
 m ;Ā‹12x
C% 7-1'2023·āĂŨũ·ö÷)ÿt8
𝑓(𝑥) = 𝑎e𝑥−𝑥2:z„Ô‹L𝑥1E𝑥2.
(1)0𝑎;Ā‹12Ž
(2)Ÿ𝑥2 ≥3𝑥1nM#$%𝑥1 +𝜆𝑥2 ≥2𝑥1𝑥2&*+M0𝜆;Šė‹.
      
      
C% 7-2'2024·IJij·ö÷)ÿt8
𝑓(𝑥) = 𝑥ln𝑥−
𝑎
2𝑥2−𝑥+ 𝑎(𝑎∈R)M𝑓′(𝑥)d𝑓(𝑥);G8
x
(1)Ü𝑎=
1
2nMŸ𝑔(𝑥) = 𝑓′(𝑥)•[[𝑡,𝑡+ 1](𝑡> 0)˜;Šm‹dℎ(𝑡)M0ℎ(𝑡)Ž
(2)ÿt𝑥1,𝑥2J8
 f'x);z„Ô‹LMÊ𝑥1 < 𝑥2MŸ#$%e1+𝑚< 𝑥1𝑥𝑚
2 &*+M0ª
 m ;Ā‹12x
C% 7-3'2023·Ģď·ö÷)ÿt8
𝑓(𝑥) =
1
2𝑚𝑥2 + (𝑚−1)𝑥−ln𝑥(𝑚∈R)M𝑔(𝑥) = 𝑥2−
1
2e𝑥+1.
(1)45𝑓(𝑥);ĕĖ~Ž
(2)Ü𝑚> 0nMŸ“·ĈU;𝑥1 ∈(0, + ∞)MŪġ•𝑥2 ∈[1, + ∞)M‚𝑓(𝑥1) ≥𝑔(𝑥2)M0𝑚;Ā‹12.
aEĕ
1x'2024·øù·÷üýþ)Ü𝑥> 0nM𝑥2 ⋅e4𝑥−2ln𝑥≥𝑎𝑥+ 1&*+MA¬
𝑎Šm‹d'    )
Ax
4
e
Bx4
Cx
4
e2
Dx8
2x'2024·øùīĬ·÷üýþ)Ÿġ•𝑥∈(0, + ∞)M‚#$%𝑎2𝑥4 +𝑥≥e𝑎𝑥2 + ln2𝑥*+MA¬
𝑎;Ā
‹12d'    )
Ax 1
2e , + ∞
Bx 1
e , + ∞
Cx −∞, 1
e
Dx −∞, 1
2e
3x'2024·Ģď·÷üýþ)ÿt𝜆> 0M“ĈU;𝑥> 1M#$%e2𝜆𝑥−lne
1
2𝜆ln𝑥≥0&*+MA¬
𝜆;Ā‹
12d'    )
      
      
Ax 1
e , + ∞
Bx 1
2e , + ∞
Cx[2e, + ∞)
Dx[e, + ∞)
4x'2024·đĒ­ū·÷üýþ)ÿt8
𝑓(𝑥) = 𝑎𝑒𝑥+1n
𝑎
𝑥+2−2MŸ𝑓(𝑥) > 0&*+MAª¬
𝑎;Ā‹12
J'    )
Ax0 < 𝑎< e
Bx𝑎> e2
Cx𝑎> e
Dx𝑎> 2e
5x'2024·IJij·÷üýþ)Ÿ·𝑥;#$%(e−1)(ln𝑥+ 𝑎𝑥) ≥𝑥e𝑎𝑥−1•𝑥∈1
2 ,1 ¤:!MAª¬
𝑎;Ā
‹12J'    )
Ax(0,2 + 2ln2]
Bx 1
e ,e
Cx(0,4]
Dx 1
2e ,e
6x'2024·IJij·÷üýþ)ÿt8
𝑓(𝑥) =
1−eln𝑥−3
𝑥
,𝑥> 0
(𝑥−2)e𝑥−2e𝑥
𝑥,𝑥< 0
 
MŸ#$%𝑓(𝑥) >
𝑎
𝑥&*+MA¬
𝑎;Ā
‹12d'    )
Ax(−∞,3)
Bx(6e−2, + ∞)
Cx(6e−2,3)
Dx(−∞,6e−2)
7x'2024·»Ŭ·÷üýþ)ÿt8
𝑓(𝑥) =
ln𝑥
𝑥,𝑔(𝑥) = 𝑎𝑥e−𝑎𝑥MŸġ•𝑥1 ∈(0,1),𝑥2 ∈(−∞,0)‚𝑓(𝑥1) = 𝑔
(𝑥2)MA¬
𝑎;Ā‹12d'    )
Ax(−∞,−2)
Bx(−2,−1)
Cx(−1, + ∞)
Dx(0, + ∞)
8x'2023·˜Īŭ=·a÷)Ÿġ•¬
𝑎, 𝑏M“ĈU¬
𝑥∈[0,1]M‚#$%𝑥3 −𝑚≤𝑎𝑥+ 𝑏≤𝑥3  +𝑚&*
+MA¬
 m ;Ā‹12J'     )
Ax
3
9 ,  + ∞
Bx 8 3
9 ,  + ∞
Cx
3
3 ,  + ∞
Dx
3
2 ,  + ∞
öEë
9x'2024·Ů·a÷)Ğ𝑓(𝑥) = (1 + 𝑥)ln𝑥,𝑔(𝑥) = (𝑎−1)𝑥MŸ𝑓(𝑥) ≤𝑔(𝑥)•𝑥∈[1,2]˜&*+MA¬
 a
;‹¦ÀJ'    )(Ś:ln2 ≈0.69)
Ax
3−ln2
2
Bx3
Cx2
Dx
2+3ln2
2
10x'2024·Ģďůć·÷üýþ)ÿt8
𝑓(𝑥) = 𝑥cos𝑥−sin𝑥MÄÅĽ5Pª«;J'    )
Ax8
𝑓(𝑥)•𝑥=
π
2nMĀ‚Ôė‹−1
Bx“·∀𝑥∈[0,π]M𝑓(𝑥) ≤0&*+
      
      
CxŸ0 < 𝑥1 < 𝑥2 < πMA
𝑥1
𝑥2 <
sin𝑥1
sin𝑥2
DxŸ“·∀𝑥∈0, π
2 M#$%𝑎<
sin𝑥
𝑥< 𝑏&*+MA𝑎;Šm‹d
2
πM𝑏;Šė‹d1
11x'2024·ĥŰ·÷üýþ)Ğ𝑥1,𝑥2(𝑥1 < 𝑥2)JĚ𝑦= 𝑎9ęĚ𝑓(𝑥) = 𝑥(1−ln𝑥);z„ZL;űŲųMA
'   )
Ax𝑥1𝑥2 < e 
Bx𝑥2ln𝑥1 > 𝑥1ln𝑥2 
Cx∃𝑎∈(0,1),𝑥2−𝑥1 > e𝑎 
Dx∀𝑎∈(0,1),𝑥1ln𝑥1 + 𝑥2 > 𝑎
ÝEŴŵ
12x'2024·āĂ*ă·Ý÷)Ÿ#$%e𝑚𝑥(𝑚𝑥−ln2)−𝑥ln𝑥2 ≥0M“ĈU𝑥∈1
e , + ∞&*+MAª¬
𝑚;
Ā‹12J      
.
13x'2024·đùŶŷ·Ý÷)Ÿ𝑥𝑘−e𝑥
e𝑥
≥2(𝑘ln𝑥−𝑥)(𝑘> 0)“ĈU;𝑥∈(0, + ∞)&*+MA k ;Ā‹12J
      
x
14x'2024·Ĺĥ·Ý÷)ÿt8
𝑓(𝑥) = (𝑥−2)e𝑥+ ln𝑥M𝑔(𝑥) = 𝑎𝑥+ 𝑏M“ĈU𝑎∈(−∞,1]Mġ•𝑥∈(0,1)
‚#$%𝑓(𝑥) ≥𝑔(𝑥)*+MAž²³;𝑏;ŠmĿ
d      
.
āE!ň
15x'2024·ĠùŸŹ·Ý÷)ÿt8
𝑓(𝑥) = 𝑥2−2𝑥+ 𝑎ln𝑥,(𝑎∈𝑅).
(1)458
;ĕĖ~Ž
(2)Ÿ“ĈU;𝑥1,𝑥2 ∈(0, + ∞),𝑥1 ≠𝑥2MÂ
𝑥2𝑓(𝑥1)−𝑥1𝑓(𝑥2)
𝑥1−𝑥2
> 0&*+MA¬
𝑎;Ā‹12.
16x'2024·IJij·÷üýþ)ÿt8
𝑓(𝑥) = 𝑥2−2𝑎ln𝑥−2(𝑎∈R)x
(1)45𝑓(𝑥);ĕĖ~Ž
(2)Ÿ#$%𝑓(𝑥) ≤2(ln𝑥)2 + 𝑥2−2𝑥•¹º(1, + ∞)˜:!M0¬
 a ;Ā‹12x
      
      
17x'2024·đĒźŻ·a÷)Ğ8
𝑓(𝑥) = e𝑥+𝑎sin𝑥M𝑥∈[0, + ∞).
(1)Ü𝑎= −1nM𝑓(𝑥) ≥𝑏𝑥+ 1•[0, + ∞)˜&*+M0¬
𝑏;Ā‹12Ž
(2)Ÿ𝑎> 0,𝑓(𝑥)•[0, + ∞)˜ġ•żLM0¬
𝑎;Ā‹12.
18x'2024·āĂČč·÷üýþ)ÿt8
𝑓(𝑥) = 𝑥2
2 −𝑥ln𝑥,𝑔(𝑥) = 𝑎(𝑥+ ln𝑥) + 𝑎2−
𝑎
2.
(1)0𝑓(𝑥)ŽžL;ěĚWXŽ
(2)0<
ġ•𝑎∈0, 1
2 M‚𝑓(𝑥) ≥𝑔(𝑥)•¹º(1, + ∞)¤&*+MÊ𝑓(𝑥) = 𝑔(𝑥)•(1, + ∞)¤:!.
19x'2024·Ĝĝ·Ý÷)ÿt𝑓(𝑥) = (2𝑥−1)e𝑎𝑥−𝑥•𝑥= 0¿;ěĚWXd𝑥+ 𝑦+ 𝑏= 0x
(1)0¬
𝑎,𝑏;‹Ž
(2)<=
𝑓(𝑥)ſ:a„Ô‹L𝑥0MÊ𝑓(𝑥0) < −
3
4x
(3)Ÿ𝑔(𝑥) = (𝑘𝑥−1)e𝑘𝑥−𝑥MJœġ•𝑘‚𝑔(𝑥) ≥−1&*+Mġ•ƀ0g𝑘;Ā‹12M#ġ•ƀƁ=Ƃx