专题3.2导数与函数的单调性七大题型（原卷版）_11.4-2_2026-11-04

      
      
专题3.2 导数与函数的单调性【七大题型】

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 F(x)= f(x)-g(x).
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 F(x)= f(x)-kx.
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 1   !
"#$%&#$'(
H 1î2024·ïð·ñòóôõ!
𝑓(𝑥) = ln(2𝑥−1)−𝑥2 +𝑥"#$¨©'(oî    õ
Aö(0,1)
Bö 1
2 ,1
Cö 1−2
2
, 1+ 2
2
Dö 1
2 , 1+ 2
2
÷: 1-1î2024·Áøùú·²ñõ!
𝑦= 𝑥ln𝑥îûûõ
Aöüý©!
Bö} 0, 1
e Áoüý©!
I} 1
e , + ∞Áoüýª!
Cöüýª!
      
      
Dö} 0, 1
e Áoüýª!
I} 1
e , + ∞Áoüý©!
÷: 1-2î2024·lm·ñòóôõþÿ!
oĀ!
É}(0, + ∞)Á#$¨ª"oî    õ
Aö𝑓(𝑥) = 3𝑥+ 2−𝑥
Bö𝑓(𝑥) =
2𝑥−2−𝑥
2𝑥+2−𝑥
Cö𝑓(𝑥) = 𝑥−𝑥3
Dö𝑓(𝑥) = log1
2 𝑥+
𝑥2 + 1
÷: 1-3î2024·āĂ㑷^ñõă›!
𝑓(𝑥)o £}𝑅Á"Ā!
IÉÜ𝑥> 0”I𝑓(𝑥) = 𝑥(1−ln𝑥)I
»Ü𝑥< 0”I𝑓(𝑥)"#$¨©'(§î    õ
Aö(−∞,−e)
Bö(−e,0)
Cö(−∞,0)
Dö(−1,0)
 2   !
"#$%
H 2î2024·ĄąĆć·²ñõă›!
𝑓(𝑥) =
1
2𝑎𝑥2−(𝑎+ 2)𝑥+ 2ln𝑥I𝑎∈Rö
(1)±Ĉĉ𝑦= 𝑓(𝑥)}𝑥= 2À"Ċĉ,𝑦ċČNI+G
𝑎"c‚
(2)‡ˆ!
𝑓(𝑥)"#$%ö
÷: 2-1î2024·āĂčĎ·ñòóôõá!
𝑓(𝑥)=−𝑎2𝑥2−𝑎𝑥+3ln𝑥−1IVW𝑎>0.
(1)‡ˆ𝑓(𝑥)"#$%‚
(2)±𝑦=𝑓(𝑥)"./,𝑥ċďĐ=I+𝑎"bcdeö
÷: 2-2î2024·đĒ·²ñõă›!
𝑓(𝑥) = 𝑥ln𝑥−e𝑥+1ö
(1)+Ĉĉ𝑦= 𝑓(𝑥)}=(1,𝑓(1))À"Ċĉåē‚
(2)‡ˆ𝑓(𝑥)}(0, + ∞)Á"#$%ö
      
      
÷: 2-3î2024·ĔĕĖė·^ñõă›!
𝑓(𝑥) = 𝑚ln𝑥+ 𝑥2−𝑥,𝑓(𝑥)"-!
§𝑓′(𝑥).
(1)‡ˆ𝑓(𝑥)"#$%‚
(2)Ü𝑚= 1”IĘę
𝑓′(𝑥+ 1) ≤
2e𝑥
𝑥+1 +
1
𝑥+1 +𝑥−1.
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H 3î2024·ĚěðĜĝĞ·ñòóôõ±!
ℎ(𝑥) = ln𝑥−
1
2𝑎𝑥2−2𝑥}[1,4]Á#$¨©I»G
𝑎"bc
de§î    õ
Aö(−∞,−1]
Bö(−∞,−1)
Cö −∞,−7
16
Dö −∞,−7
16
÷: 3-1î2024·ðĕğĠ·^ñõă›𝑎> 0IÉ𝑎≠1I±!
𝑓(𝑥) = 𝑎(ln𝑥−𝑎𝑥−1)}(1, + ∞)Á#$¨
ªI» a "bcdeoî    õ
Aö(0,
1
e]
Bö[
1
e,1)
Cö(1,e]
Dö[e, + ∞)
÷: 3-2î2024·ćġĢą·Zñõ±!
𝑓(𝑥) = log𝑎(𝑎𝑥−𝑥3)(𝑎> 0É𝑎≠1)}'((0,1)r#$¨©I»𝑎
"bcdeoî    õ
Aö[3, + ∞)
Bö(1,3]
Cö 0, 1
3
Dö 1
3 ,1
÷: 3-3î23-24 ^Á·ģĤ·ĥĦõá!
𝑓(𝑥) = 𝑎𝑥−𝑎ln𝑥(𝑎> 0É𝑎≠1)}'((1, + ∞)Á#$¨©I
»𝑎"bcdeoî    õ
Aö[e, + ∞)
Bö[e2, + ∞)
Cö[2e, + ∞)
Dö[ee, + ∞)
 4  !
,-!
./0("1
H 4î2023·úħ·ñòóôõă›!
𝑓(𝑥) =
1
6𝑥3 + sin𝑥,𝑓′(𝑥)§𝑓(𝑥)"-!
I»𝑦= 𝑓′(𝑥)"6Ĩ./
oî    õ
      
      
Aö
  
Bö
  
Cö
  
Dö
  
÷: 4-1î2024·āĂ㑷Zñõ!
𝑓(𝑥) = 2𝑎𝑥3+𝑏𝑥2+𝑐(𝑎,𝑏,𝑐∈R)"6Ĩ./ĩ.ĪīI»𝑎,𝑏,𝑐67Ĭ
ĭ§î    õ
  
Aöb<c<a
Bö𝑏< 𝑎< 𝑐
Cö𝑎< 𝑏< 𝑐
Dö𝑎< 𝑐< 𝑏
÷: 4-2î2023·ĮįĈİ·^ñõă›!
𝑓(𝑥),𝑔(𝑥)"¦C./ĩ.ĪīI»î    õ
  
Aö𝑔′(−1) < 0 < 𝑓′(−1)
Bö𝑓′(−1) < 0 < 𝑔′(−1)
Cö𝑔′(3) < 𝑓′(3)
                    
Dö𝑓′(3) < 𝑔′(3)
÷: 4-3î2024·ĤıøIJ·Zñõ!
𝑓(𝑥)o £}(−4,4)Á"ij!
IV./ĩ.ĪīI𝑓(3) = 0.á
𝑓′(𝑥)o𝑓(𝑥)"-!
I»Ô x "9:𝑓(𝑥+ 1) ⋅𝑓′(𝑥) ≥0"8¥oî    õ
      
      
Aö[0,2]
Bö[−3,0] ∪[3,4)
Cö(−5,0] ∪[2,4)
Dö(−4,0] ∪[2,3)
 5  !
#$%"2——4567
H 5î2024·ðĕğĠ·^ñõă›𝑎=
1
2 eI𝑏=
ln2
2 2I𝑐=
ln4
4 IVWe = 2.71828⋯§Ĵĵ¬
"Ķ
I»
î    õ
Aö𝑏< 𝑎< 𝑐
Bö𝑏< 𝑐< 𝑎
Cö𝑎< 𝑏< 𝑐
Dö𝑐< 𝑏< 𝑎
÷: 5-1î2024·Ĕĕúķ·ñòóôõă›𝑎= ln
6
5,𝑏=
1
6,𝑐=
1
7e
1
7I»î    õ 
Aö𝑎> 𝑏> 𝑐
Bö𝑎> 𝑐> 𝑏
Cö𝑐> 𝑏> 𝑎
Dö𝑐> 𝑎> 𝑏
÷: 5-2î2024·Ĕĕ·ñòóôõă›!
𝑓(𝑥) = 2𝑥+ 2−𝑥+ cos𝑥+ 𝑥2I±𝑎= 𝑓(5ln4π)I𝑏= 𝑓(4ln5π)I
𝑐= 𝑓(5lnπ4)I»î    õ
Aö𝑐< 𝑏< 𝑎
Bö𝑏< 𝑐< 𝑎
Cö𝑐< 𝑎< 𝑏
Dö𝑏< 𝑎< 𝑐
÷: 5-3î2024·Ĕĕúķ·ñòóôõă›𝑎=
97
98,𝑏= cos
1
7,𝑐= e−1
97I»î    õ
Aö𝑎> 𝑏> 𝑐
Bö𝑏> 𝑎> 𝑐
Cö𝑐> 𝑎> 𝑏
Dö𝑐> 𝑏> 𝑎
 6  !
#$%"2——89:
H 6î2024·ĸĕđĹ·ñòóôõă›!
𝑓(𝑥) = log4(4𝑥+1)−
1
2𝑥I±𝑓(𝑎−1) ≤𝑓(2𝑎+ 1)ãĺI»G

 a "bcde§î    õ
Aö(−∞,−2]
Bö(−∞,−2] ∪[0, + ∞)
Cö[−2,
4
3]
Dö(−∞,−2] ∪[
4
3, + ∞)
÷: 6-1î2024·ĚěðĜĝĞ·^ñõă›!
𝑓(𝑥) = (𝑥−1)3 + sin(𝑥−1) +5I»9:𝑓(2𝑥+ 1) +𝑓(1−𝑥)
≥10"8¥§î    õ
Aö[0, + ∞)
Bö[1, + ∞)
Cö[2, + ∞)
Dö[3, + ∞)
÷: 6-2î2024·ćġĻļ·^ñõá!
𝑓(𝑥)" £¤§𝑅I-
§𝑓′(𝑥)I±Ü𝑥≥0”I𝑓′(𝑥) > 2𝑥−1I
ɬÔÇÈ"G
𝑥,𝑓(−𝑥) = 𝑓(𝑥) +2𝑥I»9:𝑓(2𝑥−1)−𝑓(𝑥) < 3𝑥2−5𝑥+ 2"8¥§î    õ
      
      
Aö(−∞,1)
Bö 1
3 ,1
Cö −1
3 , + ∞
Dö −∞,−1
3 ∪(1, + ∞)
÷: 6-3î2024·Ąą·ñòóôõă›𝑓(𝑥)o £}𝑅Á"Ā!
I𝑔(𝑥) = 𝑓′(𝑥)−2e𝑥+𝑥Ľo £}𝑅Á
"Ā!
I»Ô𝑥"9:𝑔(1−𝑥2) +𝑔(2𝑥+ 2) > 0"8¥§î    õ
Aö(−∞,−1) ∪(3, + ∞)
Bö(−∞,−3) ∪(1, + ∞)
Cö(−1,3)
Dö(−3,1)
 7  -
1;<!
89:
H 7î2024·ćġľĿ·^ñõă›!
𝑓(𝑥)"-!
§𝑓′(𝑥)IÉ𝑓(1) = eIÜ𝑥> 0”I𝑓′(𝑥) <
1
𝑥+ e𝑥I
»9:
𝑓(𝑥)−ln𝑥
e𝑥
> 1"8¥§î      õ
Aö(0,1)
Bö(0, + ∞)
Cö(1, + ∞)
Dö(0,1) ∪(1, + ∞)
÷: 7-1î2024·ðŀįŁ·ñòóôõá £¤§𝑅"ij!
𝑦= 𝑓(𝑥)"-!
§𝑦= 𝑓′(𝑥)I±𝑓′(𝑥) +
(𝑥+ 1)2Ľ§ij!
IÉ𝑓(2𝑎+ 4) > 𝑓(𝑎2 + 1)I»G
𝑎"bcdeoî    õ
Aö(−∞,−1) ∪(3, + ∞)
Bö(−∞,−3) ∪(1, + ∞)
Cö(−3,1)
Dö(−1,3)
÷: 7-2î2024·łė·²ñõă›!
𝑓(𝑥)" £¤§(−∞,0)IV-!
𝑓′(𝑥)Ńń𝑥𝑓′(𝑥)−2𝑓(𝑥) > 0I
»9:𝑓(𝑥+ 2024)−(𝑥+ 2024)2𝑓(−1) < 0"8¥§î    õ
Aö(−2025,−2024)
Bö(−2024,0)
Cö(−∞,−2024)
Dö(−∞,−2025)
÷: 7-3î2024·ĔĕĖė·ñòóôõ㛠£}(0, + ∞)Á"!
𝑓(𝑥)Ńń𝑓′(𝑥)−
𝑓(𝑥)
𝑥−1 > 0IÉ𝑓(1)
= 1I»9:𝑓(e𝑥)−(𝑥+ 1)e𝑥> 0"8¥§î    õ
Aö(0, + ∞)
Bö(1, + ∞)
Cö(−∞,0)
Dö(−∞,1)
Z&#
1öî2024·āĂ㑷ñòóôõ!
𝑦=
1
2𝑥2−ln𝑥"#$¨ª'(§î    õ
Aö(−1,1]
Bö[−1,1]
Cö[1, + ∞)
Dö(0,1]
2öî2024·Áø·^ñõ}'(𝐼ÁI𝑓′(𝑥) > 0o!
𝑦= 𝑓(𝑥)}Ņ'(üý©"î    õÅÆ
AöÄCņ>
Böņ>ÄC
CöÄ>
DöŇÄCĽņ>
      
      
3öî2024·ňĤʼnŊ·ñòóôõă›𝑎= ln
7
5I𝑏= cos
2
5I𝑐=
2
5I»𝑎,𝑏,𝑐"671§î    õ
Aö𝑎> 𝑏> 𝑐
Bö𝑏> 𝑐> 𝑎
Cö𝑐> 𝑏> 𝑎
Dö𝑐> 𝑎> 𝑏
4öî2024·ŋŌ·ñòóôõă›!
𝑓(𝑥) = 𝑥𝛼(𝑥> 0)I𝛼§G
I𝑓(𝑥)"-!
§𝑓′(𝑥)I}ōZNŎŏi
1WI𝑓(𝑥),𝑓′(𝑥)"6Ĩ./ÒQoî    õ
Aö
Bö
Cö
Dö
5öî2024·ðŀŐĒ·ñòóôõ±!
𝑓(𝑥) =
1
2sin2𝑥−𝑎cos𝑥}(0,π)Á#$¨©I»𝑎"bcdeoî    õ
Aö(−∞,−1]
Bö[−1, + ∞)
Cö(−∞,1]
Dö[1, + ∞)
6öî2024·ðĕįő·^ñõă›!
𝑓(𝑥)" £¤§RIÉ𝑓(2) = −1I¬ÇÈ𝑥∈𝑅I𝑓(𝑥) + 𝑥𝑓′(𝑥) < 0I»
9:(𝑥+ 1)𝑓(𝑥+ 1) > −2"8¥oî    
õ
Aö(−∞,1)
Bö(−∞,2)
Cö(1, + ∞)
Dö(2, + ∞)
7öî2024·rŒœŔŕŖë·²ñõă›!
𝑓(𝑥) = 𝑥2−cos𝑥I»𝑓ln2
2
I𝑓−ln3
3
I𝑓−ln5
5
"671§
î    õ
Aö𝑓−ln5
5
< 𝑓−ln3
3
< 𝑓ln2
2
Bö𝑓ln2
2
< 𝑓−ln5
5
< 𝑓−ln3
3
Cö𝑓−ln5
5
< 𝑓ln2
2
< 𝑓−ln3
3
Dö𝑓−ln3
3
< 𝑓−ln5
5
< 𝑓ln2
2
8öî2024·ąŗŘĂ·^ñõ㛠£} R Á"Ā!
𝑓(𝑥)"./oZÅřŚa"ĈĉI𝑓′(𝑥)o𝑓(𝑥)"-
!
IÜ𝑥> 0”I3𝑓(𝑥) + 𝑥𝑓′(𝑥) > 0IÉ𝑓(2) = 2I»9:(𝑥+ 1)3𝑓(𝑥+ 1) > 16"8¥§î    õ
Aö(1, + ∞)
Bö(−∞,−2) ∪(2, + ∞)
Cö(−∞,1)
Dö(−∞,−3) ∪(1, + ∞)
²&X
      
      
9öî2024·ĸġśŜ·Zñõ±𝑓(𝑥) = −
1
3𝑥3 +
1
2𝑥2 +2𝑥+ 1o'((𝑚−1,𝑚+ 4)Á"#$!
I»G
𝑚"c
Òŝoî   õ
Aö−4
Bö−3
Cö3
Dö4
10öî2024·ģįĎ·ñòóôõă›𝑓(𝑥)§(0, + ∞)Á"Ò-!
IÉ(𝑥+ 1)𝑓′(𝑥) > 𝑓(𝑥)I»þÿ9:
Z ãĺ"oî    õ
Aö3𝑓(4) < 4𝑓(3) Bö4𝑓(4) > 5𝑓(3)
Cö3𝑓(3) < 4𝑓(2)
Dö3𝑓(3) > 4𝑓(2)
11öî2024·ïðŞĒ·Zñõă›𝑔(𝑥)o £¤§R"!
𝑓(𝑥)"-!
I𝑓(0) = 1I𝑓(1) = 0I𝑔(𝑥) +𝑔(2−𝑥)
= 0I
𝑓(𝑥)+𝑔(𝑥)
𝑥−1
> 0I»þÿşæ¸Ÿ"oî    õ
Aö𝑓(2) = 1
Bö𝑓(3) >
1
eîe§Ĵĵ¬
"Ķ
Ie ≈2.71828 ⋅⋅⋅õ
CöÑ}𝑥0 ∈RI𝑓(𝑥0) < 0
Dö±𝑥0 ∈(0,1)I»𝑓(𝑥0) ∈(0,1)
^&Ž
12öî2024·ģĤŠŞ·²ñõ±𝑎=
1
3I𝑏= tan
π
9I𝑐= ln
5
4I» aIbIc "671o      
îš“<”ř
Ţõö
13öî2024·āĂ·ñòóôõă›!
𝑓(𝑥) = 𝑥2 + (𝑥−2)e𝑥−2𝑥+ 5}'((3𝑚−1,𝑚+ 2)Á#$I» m "
bcdeo      
.
14öî2024·ţ·^ñõá!
𝑓(𝑥)}𝑅ÁÑ}-
𝑓′(𝑥)I¬ÔÇÈ"G
𝑥I𝑓(𝑥)−𝑓(−𝑥) +2𝑥= 0IÜ
𝑥∈(−∞,0]”I𝑓′(𝑥) +1 < 2𝑥.±𝑓(2 + 𝑚)−𝑓(−𝑚) ≤2𝑚+ 2I»G
𝑚"bcdeo      
.
ā&8
15öî2024·ŋŌ·^ñõă›!
𝑓(𝑥) =
e𝑥
𝑥+𝑎.
(1)Ü𝑎= 1”I+𝑓(𝑥)}=(0,𝑓(0))À"Ċĉåē‚
(2)±𝑓(𝑥)}'((0, + ∞)Á#$¨©I+G
 a "bcde.
      
      
16öî23-24 ^Á·Ĥı·ŤťŦŧõ㛠R ÁÒ-!
𝑓(𝑥)"./ĩ.ĪīI89:(𝑥2−2𝑥−3)𝑓′(𝑥)
> 0.
    
17öî2024·ĚěðĜĝĞ·^ñõă›!
𝑓(𝑥) = ln𝑥−
𝑥−1
𝑥+1
(1)+𝑓(𝑥)}(1,𝑓(1))À"Ċĉ‚
(2)45ln
2023
2024,−
1
4047"67Ũşęũ.
18öî2024·ćġŪū·²ñõă›!
𝑓(𝑥) = ln𝑥+ 𝑎𝑥2−𝑥+ 𝑎+ 1.
(1)ĘęĈĉ𝑦= 𝑓(𝑥)}𝑥= 1À"Ċĉ]Ŭ=‚
(2)‡ˆ𝑓(𝑥)"#$%‚
19öî2024·ćġ·²ñõă›!
𝑓(𝑥) = 𝑎2𝑥e𝑥−𝑥−ln𝑥ö
(1)Ü𝑎=
1
e”I+𝑓(𝑥)"#$'(‚
(2)Ü𝑎> 0”I𝑓(𝑥) ≥2−𝑎I+𝑎"bcdeö