专题3.3导数与函数的极值、最值七大题型（原卷版）_11.4-2_2026-11-04

      
      
专题3.3 导数与函数的极值、最值【七大题型】

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 f(x)G[a,b]À*1&*¢£¤¥
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 f(x)*É%&6 f(a)Df(b)ʗDË\1Q*¢·µ1Q&D1R*¢·µ1R&.
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ì 1+2024·íîïð·¢ñ-ò𝑎> 𝑏Dó 
𝑦= 𝑎(𝑥−𝑎)(𝑥−𝑏)2*!"ßôd+    -
A 
B 
  
C 
  
D 
  
õ| 1-1+2024·ö÷øù·úñ-() 
𝑓(𝑥) = (𝑎𝑥+ 1)e𝑥DÞ©û« 4 ·!"
      
      
Ë\Dß»üµ 
𝑓(𝑥)*QÕ!"*·
µ+    -
A 1
B 2
C 3
D 4
õ| 1-2+23-24 úû·ö÷øý·þÿ™-Ā!d𝑦= 𝑓(𝑥)*: 
𝑓′(𝑥)*!"Diāû«ö·#$D
Ë\æ*#$d+      -
A à𝑥= −1“D𝑓(𝑥)IJ%Q&
B 𝑓(𝑥)G[−2,1]Àdă 
C à𝑥= 1“D𝑓(𝑥)IJ%Q&
D 𝑓(𝑥)G[−1,2]Àdă 
DG[2,4]ÀdĄ 
õ| 1-3+2024·ąĆ·ñćĈĉ- 
𝑓(𝑥) = 𝑎𝑥𝑚(3−𝑥)𝑛GÃÄ[0,3]À*!ĊĀ!Dó mDn *&ßô
d+    -
A 𝑚= 2D𝑛= 2 B 𝑚= 2D𝑛= 1
C 𝑚= 1D𝑛= 2
D 𝑚= 1D𝑛= 1
 2  '() 
*%&
ì 2+2024·ċČ·ñćĈĉ- 
𝑓(𝑥) = (𝑥−2)e𝑥−e2𝑥−2*%R&µ+    -
A e2−2
B −2e2−2
C 2−2e2
D −2−e2
õ| 2-1+2024·čĎď÷·¢ñ-ò 
𝑓(𝑥) = (𝑥2−𝑎𝑥−2)e𝑥G𝑥= −2ÆIJ%Q&Dó𝑓(𝑥)*%R&
µ+    -
      
      
A −6e2
B −4e
C −2e2
D −e
õ| 2-2+2024·ąĆ·ñćĈĉ-() 
𝑓(𝑥) = 4𝑥e𝑥−e2𝑥−2e𝑥D𝑓′(𝑥)µ𝑓(𝑥)*: 
D𝑔(𝑥) = 𝑓′(𝑥)
e𝑥D
ó+    -
A 𝑔(𝑥)*%Q&µ4e2−2DÌ%R&
B 𝑔(𝑥)*%R&µ4e2−2DÌ%Q&
C 𝑔(𝑥)*%Q&µ4ln2−2DÌ%R&
D 𝑔(𝑥)*%R&µ4ln2−2DÌ%Q&
õ| 2-3+2024·ĐîđĒ·ñćĈĉ-() 
𝑓(𝑥)wË: 
𝑓′(𝑥)*m§¨ēµ𝑅DĔ𝑓′(𝑥)−𝑓(𝑥) = 𝑥2
e2𝑥,𝑓(0) = 0Dó𝑓(𝑥)+    -
A x¢·%R&,D¢·%Q&,
B x±·%R&,D¢·%Q&,
C 1ĕx¢·%R&,DÌ%Q&,
D 1ĕx¢·%Q&,DÌ%R&,
 3  %&+,-'.

ì 3+2024·ĖčėĘę·¢ñ-() 
𝑓(𝑥) = e𝑥−𝑎𝑥2G𝑅ÀÌ%&Dó𝑎*I&Ěěd+    -
A  −∞, e
2
B  −∞, e
2
C [0,e)
D  0, e
2
õ| 3-1+2024·ö÷Ĝĝ·ñćĈĉ-() 
𝑓(𝑥) = 𝑥3 +𝑎𝑥2 +𝑏𝑥+ 𝑎2G𝑥=−1Æx%&8Dó𝑓(1)z
ā+    -
A −4
B 16
C −4~ 16
D 16 ~ 18
õ| 3-2+2024·ĐĞğĠę·rñ-() 0 d 
𝑓(𝑥) = 𝑥3 +𝑎𝑥2 +1*%Q&,Dó𝑎*I&Ěěµ
+    -
A (−∞,0)
B (0, + ∞)
C  −∞,−2
3
D  −2
3 , + ∞
õ| 3-3+2024·ąĆ·ñćĈĉ-() 
𝑓(𝑥) =
𝑎(sin𝑥+cos𝑥)
e𝑥
+𝑥G(0,π)Àġx±·%&,DóĢ
 a *
I&Ěěd+    -
A  0,
2
2 e
π
4
B (−∞,eπ)
C (0,eπ)
D 
2
2 e
π
4, + ∞
 4  '/0. 
*1&
ì 4+2024·ģĤĤù·úñ- 
𝑓(𝑥) =
𝑥
𝑥2+1G[−3,3]À*1Q&L1R&@ĥd+   -
A 
6
13,−
6
13
B 
2
5,−
2
5
C 
3
10,−
3
10
D 
1
2,−
1
2
õ| 4-1+2024·čĎĦħ·¢ñ- 
𝑓(𝑥) = sin𝑥−(𝑥+ 2)cos𝑥−1GÃÄ[0,2π]À*1R&51Q&@ĥ
      
      
µ+    -
A −2π−3,π +1
B −2π−3,−3
C −3,π +1
D −3,2
õ| 4-2+2024·ĨĩĪī·úñ-òā x */z|e𝑥+𝑥+ 2ln
1
𝑥≥𝑚𝑥2 + ln𝑚ĬĭĮDóĢ
 m *1Q
&µ+     -
A 
1
2
B e2
4
C e2
2
D e2
õ| 4-3+2024·íî·ñćĈĉ-() 
𝑓(𝑥) =
𝑎
2𝑥2−𝑥(ln𝑥−𝑏−1),𝑎,𝑏∈𝑅DĔ𝑓(𝑥)GÃÄ(0, + ∞)À2
3įăDó2𝑎+ 𝑏*1R&µ+    -
A 0
B 
1
e
C ln2
D -1
 5  '0. 
*1&
ì 5+2024·øİøī·ñćĈĉ-() 
𝑓(𝑥) = 𝑥e𝑎𝑥+𝑎> 0-.
(1)'𝑓(𝑥)GÃÄ[−1,1]À*1Q&61R&{
(2)à𝑎≥1“D'ı
𝑓(𝑥) ≥ln𝑥+ 𝑥+ 1.
õ| 5-1+2024·IJĤijĴ·úñ-() 
𝑓(𝑥) = 𝑎ln𝑥−2𝑥−𝑎2
𝑥(𝑎≠0).
(1)à𝑎= 1“D'𝑓(𝑥)*23ÃÄL%&{
(2)'𝑓(𝑥)GÃÄ(0,1]À*1Q&.
õ| 5-2+2024·ĵĶ·ñćĈĉ-() 
𝑓(𝑥) = 𝑎𝑥−e𝑥+1 
(1)' 
𝑦= 𝑓(𝑥)*1&{
(2)ò𝑎= 3DķĸĹ𝑦= 𝑓(𝑥)6𝑥ĺĂĻĺ*ļ,µ𝑃DĽĸĹG,𝑃Æ*ľĹX}µ𝑦= 𝑔(𝑥)D'ı
∀𝑥∈R
,𝑓(𝑥) ≤𝑔(𝑥)；
      
      
õ| 5-3+2024·ģĤĤù·úñ-() 
𝑓(𝑥) = 𝑎𝑥−𝑥ln𝑥 
(1)à𝑎= 1“D'ĸĹ𝑦= 𝑓(𝑥)G(e,𝑓(e))Æ*ľĹX}{
(2)…†𝑓(𝑥)G[1,e]À*1Q&{
(3)dĿŀGĢ
 aDŁJił´𝑥> 0Dx𝑓(𝑥) ≤𝑎ŃòŀGD' a ßI*&¸ĭ*ń8{ò/ŀGDŅņ
Ň 
 6  () 
1&'.

ì 6+2024·ģĤňî·ñćĈĉ-() 
𝑓(𝑥) = 𝑥e𝑥+𝑎GÃÄ[0,1]À*1R&µ 1DóĢ
 a *&µ
+    -
A -2
B 2
C -1
D 1
õ| 6-1+2023·ö÷Ĝĝ·rñ-ò 
𝑓(𝑥) = (𝑥−𝑚)2−2,𝑥< 0
2𝑥3−3𝑥2,𝑥≥0
 
*1R&d−2DóĢ
𝑚*I&Ěě
d+    -
A 𝑚< 0
B 𝑚≤0
C 𝑚> 0
D 𝑚≥0
õ| 6-2+2024·čĎʼnŊ·ñćĈĉ-()𝑓(𝑥) = 2𝑥3−6𝑥2 +𝑚+𝑚µn
-G[−2,2]Àx1Q& 3Dó
 
𝑓(𝑥)G[−2,2]À*1R&µ+    -
A −3
B −5
C −37
D −39
õ| 6-3+2024·ĨĩŋŌ·ñćĈĉ-() 
𝑓(𝑥) = 𝑥3−𝑎𝑥2 +3𝑥G𝑅À23įăDĔ𝑔(𝑥) = 𝑥+
𝑎
2𝑥G
ÃÄ(1,2]Àōx1Q&Ŏx1R&DóĢ
𝑎*I&Ěěd+    -
A [3,4)
B (2,3]
C (3,4]
D [2,3)
 7   
2345%&61&*789
ì 7+2024·ö÷ĭ·úñ-() 
𝑓(𝑥) = (𝑥+ 𝑎)ln𝑥*: 
µ𝑓′(𝑥) 
      
      
(1)à𝑎= 1“D'𝑓′(𝑥)*1R&{
(2)ò𝑓(𝑥)ŀG±·%&,D' a *I&Ěě 
õ| 7-1+2024·ģĤŏĒ·ñćĈĉ-() 
𝑓(𝑥) = ln𝑥+ 𝑥+
2
𝑥 
(1)' 
𝑦= 𝑓(𝑥)*23ÃÄ{
(2)ķ 
ℎ(𝑥) = e𝑥
𝑥−𝑡⋅𝑓(𝑥)Dò𝑦= ℎ(𝑥)ġx±·%&,D'Ģ
𝑡*I&Ěě 
õ| 7-2+2024·ąĆ·ñćĈĉ-ķ 
𝑓(𝑥) = 𝑥2−2𝑎𝑥+ ln𝑥(𝑎∈R).
(1)à𝑎=
3
2“D'𝑓(𝑥)*%&,{
(2)à 2 < 𝑎≤
3
2“Dķ𝑥1 < 𝑥2DĔ𝑥1 ∈1
2 ,
2
2 ,𝑥2 ∈
2
2 ,1 DŐ𝑓(𝑥1)−𝑓(𝑥2)*1Q&µ𝐺(𝑎)D”'𝐺(𝑎)*I
&Ěě.
õ| 7-3+2024·ąĆ·ñćĈĉ-() 
𝑓(𝑥) = 𝑎𝑥2−(ln𝑥)2(𝑎∈𝑅) 
(1)à𝑎= 1“D…† 
𝑓(𝑥)*234 
(2)ò𝑓(𝑥)x±·%&,𝑥1,𝑥2 
①'Ģ
𝑎*I&Ěě{
②'ı
𝑥1𝑥2 > e 
      
      
¢52
1 +2024·ÀőŒœ·úñ-Ā!D()ŔĹ𝑦= 𝑘𝑥+ 𝑚6 
𝑦= 𝑓(𝑥),𝑥∈(0, + ∞)*!"kľā±,Dó
 
𝑦= 𝑓(𝑥)−𝑘𝑥x+    - 
A 2 ·%Q&,D1 ·%R&,
B 3 ·%Q&,D2 ·%R&,
C 2 ·%Q&,DÌ%R&,
D 3 ·%Q&,DÌ%R&,
2 +2024·IJİŕù·ñćĈĉ-() 
𝑓(𝑥) = e𝑥−𝑎ln𝑥GÃÄ(1,2)À23įăDó𝑎*1Q&+    -
A e2
B e−1
C e
D 2e2
3 +2024·øİŖŗ·ñćĈĉ-() 
𝑓(𝑥) =
𝑎(sin𝑥+cos𝑥)
e𝑥
+𝑥G(0,π)Àġx±·%&,DóĢ
𝑎*I&
Ěěd+ŘŘ-
A  0, e
π
2
2
B  0,
2
2 e
π
4
C  e
π
2
2 , + ∞
D 
2
2 e
π
4, + ∞
4 +2024·ģĤùř·ñćĈĉ-HŚ¸śZÙ¢·ŜŝŝŞ.›şŠDGŜŝšŢţ:û;Ť¢·ƒ
ťµ27cm3*ŦŧľŨĭ¢·ŦũDľŨÓ}\ŪūŬ¿/>DóŦũƒť*1Q&µ+    -
A 4cm3
B 8cm3
C 12cm3
D 16cm3
5 +2024·ŭŮČůůŰű·úñ-() 
𝑓(𝑥) = e2𝑥+ (𝑥−2𝑎)e𝑥+ 𝑎2(𝑎∈R)*1R&µ𝑔(𝑎)Dó𝑔(𝑎)*
1R&µ+    -
A −e
B −
1
e
C 0
D 1
6 +2024·ŲųŴī·¢ñ-()𝑥1,𝑥2Dd 
𝑓(𝑥) = (𝑥−1)3−𝑥±·%&,Dó+    -
A 𝑥1 + 𝑥2 = −2
B 𝑥1 + 𝑥2 = 1
C 𝑓(𝑥1) +𝑓(𝑥2) = −2D 𝑓(𝑥1) +𝑓(𝑥2) = 2
7 +2024·øİŖŗ·úñ-ķ 
𝑓(𝑥) = 𝑥+ e𝑥D𝑔(𝑥) = 𝑥+ ln𝑥DòŀG𝑥1D𝑥2DŁJ𝑓(𝑥1) = 𝑔(𝑥2)Dó|𝑥1−𝑥2|
*1R&µ+    -
      
      
A 
1
e
B 1
C 2
D e
8 +2024·ĞŵŶ§·rñ-Uª
)žFy^dŷŸ™
*¢·ĵ;Ź*DźŸUŷ
ٻ
)žDŻW 
𝑓(𝑥) = 𝑥𝑥D𝑥∈(0, + ∞)Dóû«ż/æ*d+    -
A 𝑓(𝑥)xĔŽx¢·%&,
B 𝑓(𝑥)G 1
e + ∞À23žă
C ŀGĢ
𝑎∈(0, + ∞)DŁJ𝑓(𝑎) =
1
e
D 𝑓(𝑥)x1R&
1
e
1
e
ú5ĕ
9 +2024·ĵĶ·rñ-ò 
𝑓(𝑥) = 𝑎ln𝑥−2𝑥2 +𝑏𝑥ōx%R&Ŏx%Q&Dó+    
-
A 𝑎𝑏< 0
B 𝑎< 0
C 𝑏2 +16𝑎> 0
D |𝑎−𝑏| < 4
10 +2024·ċČſī·rñ-() 
𝑓(𝑥) = (𝑥+ 1)e𝑥Dóû«€†Ă¦*d+    -
A 𝑓(𝑥)GÃÄ(−2, + ∞)À23įă
B 𝑓(𝑥)*1R&µ−
1
e2
C X}𝑓(𝑥) = 2*Fx 2 ·
D : 
𝑓′(𝑥)*%&,µ−3
11 +2024·ŭŮČ·rñ-() 
𝑓(𝑥) = e|𝑥|cos𝑥Dóû«€†Ă¦*d+    -
A 𝑓(𝑥)*!"G, π
2 ,𝑓π
2
Æ*ľĹG y ĺÀ*ƀƁµ
π
2e
π
2
B 𝑓(𝑥)G −π
4 ,0 Àµă 
C 𝑓(𝑥)G −π
2 , π
2 À*1Q&µ 2
2 e
π
4
D ò𝑓(𝑥)G(−𝑚,𝑚)gġx 11 ·%&,DóĢ
 m *I&Ěěµ 17π
4 , 21π
4
r5Œ
12 +2024·Àő·rñ-ò 
𝑓(𝑥) = −4𝑥3 +3𝑥G(𝑎,𝑎+ 2)ÀŀG1R&DóĢ
 a *I&Ěěd      
 
13 +2024·ö÷ĭ·ñćĈĉ-ò 
𝑓(𝑥) = (𝑥−𝑎)𝑥+ ln𝑥G(0, + ∞)ÀÌ%&,Dó𝑎*I&Ěěµ      
.
14 +2023·ö÷Ƃč·ñćĈĉ-() 
𝑓(𝑥) =
1
e|ln𝑥| ,𝑥∈(0, + ∞)
ln(1−𝑥),𝑥∈(−∞,0]
 
Dóû«ż\æ*x      
 
① 
𝑓(𝑥)x±·%&,{
②òā x *X}𝑓(𝑥) = 𝑡ġx 1 ·FDó𝑡> 1{
③ 
𝑓(𝑥)*!Ċ6ŔĹ𝑥+ 𝑦+ 𝑐= 0(𝑐∈R)xĔÏx¢·ļ,{
④ò𝑓(𝑥1) = 𝑓(𝑥2) = 𝑓(𝑥3)DĔ𝑥1 < 𝑥2 < 𝑥3Dó(1−𝑥1)(𝑥2 + 𝑥3)Ì1& 
ö5FŽ
15 +2024·øİİƃ·ñćĈĉ-() 
𝑓(𝑥) =
1
2𝑥2 + (1−𝑎)𝑥−𝑎ln𝑥(𝑎∈R) 
      
      
(1)' 
𝑓(𝑥)*23ÃÄ{
(2)à𝑎> 0“D' 
𝑓(𝑥)GÃÄ[1,e]À*1Q& 
16 +2024·ĐĞƄƅƆ·rñ-() 
𝑓(𝑥) =
1
2𝑥2−(𝑎+ 1)𝑥+ 𝑎ln𝑥(𝑎> 0) 
(1)…† 
𝑓(𝑥)*234{
(2)à𝑎= 2“Dò 
𝑔(𝑥) = 𝑓(𝑥)−
1
2𝑥2 + e𝑥−1D' 
𝑔(𝑥)%&,*·
 
17 +2024·ģĤĤù·ñćĈĉ-() 
𝑓(𝑥) = 𝑎𝑥−ln𝑥−𝑎Dò𝑓(𝑥)*1R&µ 0D
(1)'𝑎*&;
(2)ò𝑔(𝑥) = 𝑥𝑓(𝑥)Dıņ:𝑔(𝑥)ŀGƇ¢*%Q&,𝑥0DĔ𝑔(𝑥0) <
1
4.
18 +2024·ģĤùř·ñćĈĉ-() 
𝑓(𝑥) = 𝑥2e−𝑥−𝑎(𝑥2−4𝑥)Gm§¨(0, + ∞)ÀÏx 1 ·%Q&,.
(1)'𝑎*I&Ěě{
(2)ò𝑓(𝑥1) = 𝑓(𝑥2),1 < 𝑥1 < 2 < 𝑥2Dıņ
𝑥1 + 𝑥2 > 4.
19 +2024·ƈƉĐĞ·úñ-()𝑎> 0D 
𝑓(𝑥) = 𝑎ln𝑥−ln(𝑥+ 1).
      
      
(1)à𝑎= 1“D'ĸĹ𝑦= 𝑓(𝑥)G,(1,𝑓(1))Æ*ľĹX}{
(2)à0 < 𝑎< 1“.
+ⅰ-'𝑓(𝑥)*23ÃÄL%&{
+ⅱ-ķ𝑓(𝑥)*%Q&µ𝑔(𝑎)D'𝑔(𝑎)*1R&{
(3)ķ𝑛∈N+DĔ𝑛≥2D'ı
 1
𝑛
2
𝑛
2 3
𝑛
3
⋯𝑛−1
𝑛
𝑛−1
> 2−𝑛2
2 .