重难点07双变量问题九大题型（原卷版）_11.4-2_2026-11-04

      
      
重难点07 双变量问题【九大题型】

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 1   !"#$
× 1(23-24 ‡Ø·ÙÚJÛ·ÜC,yiŠ
𝑓(𝑥) = 𝑥ln𝑥−
1
2𝑎𝑥2，𝑎∈Rk
(1)¢𝑔(𝑥) =
𝑓(𝑥)
𝑥IgŠ
𝑔(𝑥)MÝÞ[1,e]ß%&c'†
(2)¢àÁáÌ%𝑥1，𝑥2 ∈(0, + ∞)I𝑥1 ≠𝑥2Iâ2
𝑓(𝑥1)−𝑓(𝑥2)
𝑥1−𝑥2
< 1Iãä
𝑎%)'*+k
9 1-1(23-24 ‡Ø·ßå·Üæ,yi𝑓(𝑥) = 2𝑥+ 1−𝑥ln𝑥.
(1)gç=𝑦= 𝑓(𝑥)M𝑥= 1è%<=u·†
(2)¢àáÌ%𝑥1,𝑥2 ∈(0, + ∞)Ié𝑥1 ≠𝑥2Iâ2
𝑓(𝑥1)−𝑓(𝑥2)
𝑥1−𝑥2
> 𝑚(𝑥1 + 𝑥2)Igä
𝑚%)'*+.
9 1-2(23-24 ‡Ø·ßå·Üæ,yiŠ
𝑓(𝑥) = 𝑎𝑥2−(𝑎+ 2)𝑥+ ln𝑥
(1)ê𝑎= 1ëIgŠ
𝑓(𝑥)%0c'†
(2)¢
𝑓(𝑥1)−𝑓(𝑥2)
𝑥1−𝑥2
> −2àW<0 < 𝑥1 < 𝑥2â56Igä
𝑎%)'*+.
      
      
9 1-3(2024·ìíî﷑Æ,yiŠ
𝑓(𝑥) = 𝑥2−2𝑥+ 𝑎ln𝑥,(𝑎∈𝑅).
(1)ðñŠ
%!"#†
(2)¢àáÌ%𝑥1,𝑥2 ∈(0, + ∞),𝑥1 ≠𝑥2Iò
𝑥2𝑓(𝑥1)−𝑥1𝑓(𝑥2)
𝑥1−𝑥2
> 0456Iãä
𝑎%)'*+.
 2   %&'()'*+,$
× 2(2024·óôÏõ·Æö÷ø,yiŠ
𝑓(𝑥) = ln𝑥−𝑎𝑥−
2
𝑎𝑥.
(1,¢𝑓(𝑥) ≥0Igä
𝑎%)'*+†
(2,¢𝑔(𝑥) = 𝑓(𝑥) + 𝑥2 +
2
𝑎𝑥2¦¸0'1Êù.𝑥1I𝑥2(𝑥1 < 𝑥2)Ig2𝑔(𝑥1)−𝑔(𝑥2)%&ú'.
9 2-1(2023·ûüýþ·WÆ,yiÁ𝑥%u·𝑎𝑥−ln𝑥= 02¦¸7¨8%ÿäµ𝑥1Ā𝑥2Ié𝑥1 <
𝑥2.
(1)gä
𝑎%)'*+†
(2)ā𝑘.J
Iê𝑎ëI¢𝑥𝑘
1𝑥22&ú'eeIgJ
𝑘%'.
9 2-2(2024·ÐĂ5â·Æö÷ø,yiŠ
𝑓(𝑥) =
sin𝑥
e𝑥−𝑚,𝑥∈(0,π).
(1)gŠ
𝑓(𝑥)%!"Ýކ
      
      
(2)¢𝑥1 < 𝑥2I€𝑓(𝑥1) = 𝑓(𝑥2) = 0.
(ⅰ,g𝑚%)'*+†
(ⅱ,:;
𝑥1 + 𝑥2 < π.
9 2-3(2024·ÐĂăÛ·WÆ,yiŠ
𝑓(𝑥) = 𝑎𝑥+ 1−𝑥ln𝑥%ĄąM𝑥= 1è%<=/Ć=𝑥−𝑦= 0ć
Äk
(1)gŠ
𝑓(𝑥)%!"Ýކ
(2)¢∀𝑥1,𝑥2 ∈(0, + ∞)Ié𝑥1 > 𝑥2ëI𝑓(𝑥1)−𝑓(𝑥2) > 𝑚𝑥2
1−𝑥2
2 Igä
 m %)'*+k
 3   $-.! $
× 3(2024·Ĉĉ·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑎𝑥−
ln𝑥
𝑥I𝑎> 0k
(1)¢𝑓(𝑥)ĊM31Ig a %)'*+†
(2)¢𝑥1I𝑥2.𝑓(𝑥)%31Ié𝑥1 < 𝑥2I:;
𝑎(𝑥1 + 𝑥2)2 > 2k
9 3-1(2024·ċČč췇Æ,yi𝑓(𝑥) = −
1
2e2𝑥+4e𝑥−𝑎𝑥−5.
(1)ê𝑎= 3ëIg𝑓(𝑥)%!"Ýކ
(2)¢𝑓(𝑥)2¦¸0'1𝑥1I𝑥2I:;
𝑓(𝑥1) +𝑓(𝑥2) + 𝑥1 + 𝑥2 < 0.
      
      
9 3-2(2024·ÐĂ·Æö÷ø,yiŠ
𝑓(𝑥) = (𝑎+ 1)e𝑥−
1
2𝑥2 +1(𝑎∈𝑅)k
(1)ê𝑎= 1ëIgç=𝑦= 𝑓(𝑥)M1(0,𝑓(0))è%<=u·†
(2)ā𝑥1,𝑥2(𝑥1 < 𝑥2)BŠ
𝑦= 𝑓′(𝑥)%¦¸31Ig:
𝑥1 + 𝑥2 > 2k
9 3-3(2024·óôĎď·WÆ,yiŠ
𝑓(𝑥) = 3ln𝑥−𝑎𝑥k
(1)ðñ𝑓(𝑥)%!"#k
(2)yi𝑥1,𝑥2BŠ
𝑓(𝑥)%¦¸31(𝑥1 < 𝑥2)k
(ⅰ,gä
𝑎%)'*+k
(ⅱ,𝜆∈0, 1
2 ,𝑓′(𝑥)B𝑓(𝑥)%AŠ
k:;
𝑓′[𝜆𝑥1 +(1−𝜆)𝑥2] < 0k
 4  /0'12% $
× 4(2024·ÐĂĐđ·‡Æ,yiŠ
𝑓(𝑥) = 𝑎e𝑥−𝑥3(𝑎∈R)2‘¸0'1𝑥1,𝑥2,𝑥3 (𝑥1 < 𝑥2 < 𝑥3).
(1)gä
𝑎%)'*+†
(2)¢𝑥3 ≥2𝑥2Igä
𝑎%)'*+.
9 4-1(2023·Ĉĉ·Æö÷ø,yiŠ
𝑓(𝑥) = (𝑥+ 𝑡)ln(𝑥+ 𝑡) + (𝑡−1)𝑥(𝑡∈R)k
(1)ê𝑡= 0ëIðñŠ
𝑓(𝑥)%0'†
      
      
(2)yi𝐹(𝑥) = 𝑓(𝑥)−e𝑥IŠ
𝐹(𝑥)ĊM¦¸0'1𝑥1I𝑥2I:;
𝑥1 + 𝑥2 < 0k
9 4-2(2023·ÐĂĒēĔ·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑎e𝑥−𝑥(𝑎∈𝑅)k
(1)ê𝑎= 1ëIg𝑓(𝑥)%!"Ýކ
(2)āŠ
𝑔(𝑥) = (𝑥2−1)e𝑥−𝑥−𝑓(𝑥)Iê𝑔(𝑥)2¦¸0'1𝑥1,𝑥2(𝑥1 < 𝑥2)ëIĕ2𝑡g(𝑥2) ≥(2 + 𝑥1)
e𝑥2 + 𝑥2
2−3 56Igä
𝑡%'k
9 4-3(2023·ßåĖÙ·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑎𝑥−𝑎ln𝑥−e𝑥
𝑥.
(1)¢𝑎= 0IgŠ
𝑦= 𝑓(𝑥)%0'1†
(2)¢789𝑓(𝑥) < 0456Igä
 a %)'*+†
(3)¢Š
𝑦= 𝑓(𝑥)2‘¸7%0'1𝑥1@𝑥2@𝑥3Ié𝑓(𝑥1) + 𝑓(𝑥2) + 𝑓(𝑥3) ≤3e2−eIgä
 a %)'*
+.
 5  /312% $
× 5(2024·ÐĂ·WÆ,yiŠ
𝑓(𝑥) = 𝑎𝑥2 +𝑥−ln𝑥−𝑎k
(1)¢𝑎= 1Ig𝑓(𝑥)%&ú'†
(2)¢𝑓(𝑥)2 2 ¸31𝑥1,𝑥2I:;
𝑎(𝑥1 + 𝑥2)2 + (𝑥1 + 𝑥2) > 2k
      
      
9 5-1(2023·ėĐ·Æö÷ø,yiŠ
𝑓(𝑥) =
𝑥
e𝑎𝑥+ ln𝑥−𝑎𝑥ICe.Ę¿à
%ę
.
(1)ê𝑎= 1ëIg𝑓(𝑥)%!"Ýކ
(2)¢Š
𝑔(𝑥) = 𝑓(𝑥)−
𝑥
e𝑎𝑥2¦¸31𝑥1,𝑥2(𝑥1 < 𝑥2)I:;
𝑥1 ⋅𝑥2 > e2.
9 5-2(2023·ĚÛĚď·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑎𝑥−ln𝑥2¦¸31𝑥1,𝑥2.
(1)g𝑎%)'*+†
(2)g:
𝑥1 + 𝑥2 > 2e(CeBĘ¿à
%ę
,.
9 5-3(2023·åĐåě·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑥e𝑥+2.
(1)g𝑓(𝑥)%&ú'†
(2)ā𝐹(𝑥) = 𝑓(𝑥) + 𝑎(𝑥+ 1)2(𝑎> 0).
(ⅰ,:;
𝐹(𝑥)ĊM¦¸31𝑥1I𝑥2†
(ⅱ,:;
𝐹(𝑥)%¦¸31𝑥1I𝑥2€𝑥1 + 𝑥2 +2 < 0.
 6   %456$
× 6(2023·ÐĂĘĜ·‡Æ,yiŠ
𝑓(𝑥) = 𝑎e𝑥−𝑥22¦¸0'1𝑥1@𝑥2.
(1)g𝑎%)'*+†
(2)¢𝑥2 ≥3𝑥1ëI789𝑥1 +𝜆𝑥2 ≥2𝑥1𝑥2456Ig𝜆%&ú'.
      
      
9 6-1(2023·ėĐ·‡Æ,yiŠ
𝑓(𝑥) =
1
2𝑚𝑥2 + (𝑚−1)𝑥−ln𝑥(𝑚∈R)I𝑔(𝑥) = 𝑥2−
1
2e𝑥+1.
(1)ðñ𝑓(𝑥)%!"#†
(2)ê𝑚> 0ëI¢àÁáÌ%𝑥1 ∈(0, + ∞)IĕĊM𝑥2 ∈[1, + ∞)Iò˜𝑓(𝑥1) ≥𝑔(𝑥2)Ig𝑚%)'*+.
9 6-2(2023·Ĉĉ·‡Æ,yiŠ
𝑓(𝑥) = 𝑥ln𝑥−
𝑎
2𝑥2−𝑥+ 𝑎(𝑎∈R)I𝑓′(𝑥).𝑓(𝑥)%AŠ
k
(1)ê𝑎=
1
2ëI¢𝑔(𝑥) = 𝑓′(𝑥)M[[𝑡,𝑡+ 1](𝑡> 0)ß%&c'.ℎ(𝑡)Igℎ(𝑡)†
(2)yi𝑥1,𝑥2BŠ
 f(x,%¦¸0'1Ié𝑥1 < 𝑥2I¢789e1+𝑚< 𝑥1𝑥𝑚
2 456Igÿ
 m %)'*+k
9 6-3(2023·óôĝĐ·WÆ,yi𝑓(𝑥) = 𝑎ln𝑥+ 𝑥2¦¸7%31𝑥1,𝑥2(0 < 𝑥1 < 𝑥2).
(1)gä
 a %)'*+†
(2)¢𝑥0 =
𝑥1+𝜆𝑥2
1+𝜆(𝜆≠−1)Ié𝑓′(𝑥0) > 0456Igä
𝜆%*+.
 7   %789:;$
× 7(2023·óôĞó·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑥2 +2cos𝑥, 𝑓′(𝑥).Š
𝑓(𝑥)%AŠ
k
(1)ðñŠ
𝑓(𝑥)%!"#†
      
      
(2)yiŠ
𝑔(𝑥) = 𝑓′(𝑥)−5𝑥+ 5𝑎ln𝑥IĊM𝑔(𝑥1) = 𝑔(𝑥2)(𝑥1 ≠𝑥2)I:;
𝑥1 + 𝑥2 > 2𝑎k
9 7-1(2024·ûĐğĠ·‘Æ,yiŠ
𝑓(𝑥) = 𝑥−
1
2sin𝑥−
𝑚
2ln𝑥+ 1.
(1,ê𝑚= 2ëIġŽŠ
𝑓(𝑥)M(𝜋, + ∞)ß%!"#†
(2,ĊM𝑥1,𝑥2 ∈(0, + ∞)I𝑥1 ≠𝑥2I𝑓(𝑥1) = 𝑓(𝑥2)Ig:
𝑥1𝑥2 < 𝑚2.
9 7-2(2023·ĢģĤĥ·‡Æ,yiŠ
𝑓(𝑥) = ln𝑥I𝑔(𝑥) = 𝑥−
2
𝑥k
(1)¢𝑥0€𝑓(𝑥0) =
𝑥0+1
𝑥0−1I:;
ç=𝑦= 𝑓(𝑥)M1𝐴(𝑥0,ln𝑥0)è%<=ĦBç=𝑦= e𝑥%<=†
(2)¢𝐹(𝑥) = 𝑓(𝑥)−𝑔(𝑥)Ié𝐹′(𝑥1) = 𝐹′(𝑥2)(𝑥1 ≠𝑥2)I:;
𝐹(𝑥1) +𝐹(𝑥2) < 4ln2−7k
9 7-3(2024·ħĨėí·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑘ln𝑥+
1
e𝑥(𝑘∈𝑅).
(1)¢Š
𝑦= 𝑓(𝑥).ĩŠ
Ig𝑘%)'*+†
(2)yi0 < 𝑥1 < 𝑥2.
(i,:;
e
e𝑥2−
e
e𝑥1 > −ln
𝑥2
𝑥1 > 1−
𝑥2
𝑥1†
(ii,¢
𝑥1
e𝑥1 =
𝑥2
e𝑥2 = 𝑘I:;
|𝑓(𝑥1)−𝑓(𝑥2)| < 1.
      
      
 8  /<=2% $
× 8(2023·ÐĂ5â·Æö÷ø,¢Š
𝑓(𝑥) = 𝑎ln𝑥−
1
2𝑥2 +𝑎+
1
2(𝑥> 0)2¦¸31𝑥1,𝑥2Ié𝑥1 < 𝑥2k
(1)g a %)'*+†
(2)¢𝑓(𝑥)M(𝑥1,0)Ā(𝑥2,0)è%<=ĪÁ1(𝑥3,𝑦3)Ig:
2𝑥3 < 𝑥1 + 𝑥2k
9 8-1(2024·ċČ·‡Æ,yi𝑓(𝑥) =
1
2𝑎𝑥2 + (1−2𝑎)𝑥−2ln𝑥,𝑎> 0.
(1)g𝑓(𝑥)%!"Ýކ
(2)Š
𝑓(𝑥)%ĄīßBĬĊM¦1𝐴(𝑥1,𝑦1),𝐵(𝑥2,𝑦2)(C𝑥1 ≠𝑥2,Iò˜Ć=𝐴𝐵/Š
𝑓(𝑥)%ĄīM𝑥0 =
𝑥1+𝑥2
2 è%<=ćÄĭ¢ĊMIĮg^Ć=𝐴𝐵†¢7ĊMIĮį;x.
9 8-2(2024·ÐĂİı·‘Æ,yiŠ
𝑓(𝑥) = 𝑚+ 1
𝑚ln𝑥+
1
𝑥−𝑥I(CJ
𝑚> 0,
(1)ê𝑚= 2ëIg𝑓(𝑥)%0c'†
(2)ê𝑚∈[3, + ∞)ëIç=𝑦= 𝑓(𝑥)ßĕĊM¨IJ¦1𝑃(𝑥1,𝑓(𝑥1))@𝑄(𝑥2,𝑓(𝑥2))Iò˜ç=𝑦= 𝑓(𝑥)M1𝑃@𝑄
è%<=ij¨ćÄIg𝑥1 + 𝑥2%)'*+k
9 8-3(2024·DĴ·Æö÷ø,ĵĶMķĸ
vĹWĺCI¡^Ļ©
u·%Wļ
'Zv——ĵĶ
vkĽ­ľvĿØ
ĿĄIā r B𝑓(𝑥) = 0%µIŀŁ)𝑥0Y. r %łŃńŅ'I¢𝑓(𝑥)M1(𝑥0,𝑓(𝑥0))è
%<=/𝑥]¨ĪÁ1(𝑥1,0)Iņ𝑥1B r %W¾ńŅ'†𝑥1Ň©𝑥0Dňßʼn%¥·I˜–𝑥2Iņ𝑥2B r %‡
¾ńŅ'†WĆDňI—˜–WŊ
𝑥0,𝑥1,𝑥2,⋯,𝑥𝑛,⋯kMW>ŋ¹aØIÐŌÑ|v)'Iê𝑥𝑛−1,𝑥𝑛(𝑛∈N∗)
      
      
ńŅ'¨8ëI«'wY.Š
𝑓(𝑥)%W¸31𝑟k
(1)¢𝑓(𝑥) = 𝑥3 +3𝑥2 +𝑥−3Iê𝑥0 = 0ëIgu·𝑓(𝑥) = 0%‡¾ńŅ'(ōŎ–ú
1À¦Q,†
(2)ĵĶvCŏ…Ļ“VĆ©ç”%
ÓoIĆ=JJ).ç=%<=»Ő=IgŠ
𝑔(𝑥) = e𝑥−3M1
(2,𝑔(2))è%<=Iƒ:;
ln3 < 1 +
3
e2†
(3)¢ℎ(𝑥) = 𝑥(1−ln𝑥)I¢Á𝑥%u·ℎ(𝑥) = 𝑎%¦¸µÊù.𝑥1,𝑥2(𝑥1 < 𝑥2)I:;
𝑥2−𝑥1 > e−e𝑎k
 9   %>?$
× 9(2024·őÙŒœ·‘Æ,¢Š
𝛼(𝑥)2é³2W¸0'1𝑚IŠ
𝛽(𝑥)2é³2W¸0'1𝑛Ié
𝑚> 𝑛Iãņ𝛼(𝑥)/𝛽(𝑥)Ľ2#Ŕ𝛼−𝛽//𝑚> 𝑛k
(1)Š
𝜑1(𝑥) = sin𝑥−𝑥2/𝜑2(𝑥) = e𝑥−𝑥BĬĽ2#Ŕ𝜑1−𝜑2//𝑥0 > 0ĭƒį;xk
(2)yiŠ
𝑓(𝑥) = 𝑎e𝑥−ln(𝑥+ 1)/𝑔(𝑥) = ln(𝑥+ 𝑎)−e𝑥+1Ľ2#Ŕ𝑓−𝑔//𝑥1 > 𝑥2k
(i,g𝑎%)'*+†
(ii,:;
|𝑔(𝑥1)| > |𝑥2|k
9 9-1(2023·ûü·‡Æ,ā𝑓(𝑥)B>?MÝÞ(1, + ∞)ß%Š
IAŠ
.𝑓′(𝑥)kĿšĊMä
 a
ĀŠ
ℎ(𝑥)ICℎ(𝑥)àáÌ%𝑥∈(1, + ∞)â2ℎ(𝑥) > 0Iò˜𝑓′(𝑥) = ℎ(𝑥)(𝑥2−𝑎𝑥+ 1)IãņŠ
𝑓(𝑥)Ľ
2#Ŕ𝑃(𝑎)k
      
      
(1)āŠ
𝑓(𝑥) = ln𝑥+
𝑏+2
𝑥+1(𝑥> 1)IC b .ä
k
(i,g:
Š
𝑓(𝑥)Ľ2#Ŕ𝑃(𝑏)†
(ii,gŠ
𝑓(𝑥)%!"ÝÞk
(2)yiŠ
𝑔(𝑥)Ľ2#Ŕ𝑃(2).¡>𝑥1,𝑥2 ∈(1, + ∞)I𝑥1 < 𝑥2Iā m .ä
I
𝛼= 𝑚𝑥1 +(1−𝑚)𝑥2I𝛽= (1−𝑚)𝑥1 +𝑚𝑥2Ié𝛼> 1I𝛽> 1I¢|𝑔(𝛼)−𝑔(𝛽)| < |𝑔(𝑥1)−𝑔(𝑥2)|Ig m %)'
*+k
9 9-2(2024·őÙŕÛ·‡Æ,ĿĄIàÁç=ΓIĊMŖ𝐶€ĿØz{
       
①Ŗ𝐶/ç=Γ2ŗ1𝐴IéŖŘMç=Γř%WŚ†
②Ŗ𝐶/ç=ΓM1𝐴è2¨%<=†
③ç=Γ%AŠ
M1𝐴è%A
(wç=Γ%‡śA
,8ÁŖ𝐶M1𝐴è%‡śA
(yiŖ(𝑥−𝑎)2 +
(𝑦−𝑏)2 = 𝑟2M1𝐴(𝑥0,𝑦0)è%‡śA
8Á
𝑟2
(𝑏−𝑦0)3,†
ãņŖ𝐶.ç=ΓM𝐴1è%çŜŖIŝŞ𝑟ņ.çŜŝŞk
(1)gş=𝑦= 𝑥2MŠ1%çŜŖ%u·†
(2)gç=𝑦=
1
𝑥%çŜŝŞ%&ú'†
(3)¢ç=𝑦= e𝑥M(𝑥1,e𝑥1)Ā(𝑥2,e𝑥2)(𝑥1 ≠𝑥2)è2¨%çŜŝŞIg:
𝑥1 + 𝑥2 < −ln2k
      
      
9 9-3(2023·ßåšŢ·‡Æ,yiJ
𝑘.ţ3¬
I¢Š
𝑦= 𝑓(𝑥)I𝑥∈[0,1]€
àáÌ𝑥1,𝑥2 ∈
[0,1]I|𝑓(𝑥1)−𝑓(𝑥2)| ≤|(𝑥1 + 1)𝑘−(𝑥2 + 1)𝑘|IãņŠ
𝑦= 𝑓(𝑥).𝐿(𝑘)Š
.
(1)Š
𝑦= 2𝑥I𝑥∈[0,1]BĬ.𝐿(2)Š
﹖Įį;x†
(2)¢𝑦= 𝑓(𝑥).𝐿(1)Š
IĄąM𝑥∈[0,1]BWzŤť%ç=I𝑓(0) = 0I𝑓(1) =
1
2Ié𝑓(𝑥)MÝÞ(0,1)ß³
ĊMW¸0'1IÊùŦ𝑓(𝑥)max@𝑓(𝑥)min.Š
𝑦= 𝑓(𝑥)%&c@ú'Ig𝑓(𝑥)max−𝑓(𝑥)min%)'*+†
(3)¢𝑎> 0I𝑓(𝑥) = 0.05𝑥2 +0.1𝑥+ 𝑎ln(𝑥+ 1)Ié𝑦= 𝑓(𝑥).𝐿(−1)Š
I𝑔(𝑥) = 𝑓′(𝑥)IàáÌ𝑥,𝑦∈[0,1]I
42|𝑔(𝑥)−𝑔(𝑦)| ≤𝑀IŦ𝑀%&ú'.𝑀(𝑎)Ig𝑎%)'*+T𝑀(𝑎)Á𝑎%ŧŨ9.
W@!
1k(2023·ũŪğū·Æö÷ø,yi aIb €e𝑎= −𝑎e−2I𝑏(ln𝑏−2) = e4IC e BĘ¿à
%ę
I
ã ab %'.(    ,
Ak−e
Bk−e2
Ck−e3
Dk−e4
2k(23-24 ‘ß·ċČÙŬ·śŭŮů,yi𝑓(𝑥) = 𝑎ln𝑥+
1
2𝑥2(𝑎> 0)¢àÁá̦¸78%ÿä
𝑥1@𝑥2I
â2
𝑓(𝑥1)−𝑓(𝑥2)
𝑥1−𝑥2
> 2456Iã𝑎%)'*+B(    ,
Ak(0,1]
Bk[1, + ∞)
Ck(0,3]
Dk[1,2e)
3k(23-24 ‡Ø·ĢģĢÛ·Üæ,yi𝑥,𝑦.ÿä
Iln𝑥+ ln𝑦=
1
𝑦−𝑥Iã(    ,
Ak𝑥 ＞ 𝑦
Bk𝑥 ＜ 𝑦
Ck𝑥+ 𝑦 ＞ 1
Dk𝑥+ 𝑦 ＜ 1
4k(2023·ÙÚФ·Æö÷ø,yiĆ=𝑦= 𝑘𝑥+ 𝑡/Š
𝑦= 𝐴sin(𝜔𝑥+ 𝜑)(𝐴> 0,𝜔> 0)%ĄīŰ2¦¸
<1Iā€z{% k ”2—£)'C&c%¦¸'Êù.𝑘1Ā𝑘2Ié𝑘1 > 𝑘2Iã(    ,
Ak
3
5 <
𝑘1
𝑘2 <
5
7
Bk
7
5 <
𝑘1
𝑘2 <
5
3
Ck
5
3 <
𝑘1
𝑘2 <
7
3
Dk
7
5 <
𝑘1
𝑘2 <
7
3
5k(23-24 ‡Ø·ĢģĢÛ·ÜC,yiŠ
𝑓(𝑥) = (𝑥−2)e𝑥I¢𝑓(𝑥1) = 𝑓(𝑥2)Ié𝑥1 ≠𝑥2I𝑥1 ⋅𝑥2 > 0Iã
(    ,
Ak𝑥1 >
1
2
Bk𝑥2 <
3
2
Ck𝑥1𝑥2 > 1
Dk𝑥1 + 𝑥2 < 2
      
      
6k(2024·ÐĂ5â·WÆ,yi𝑎> 𝑏Iée𝑎−𝑎= e𝑏−𝑏= 1.01IãØŊįvÿ¹%2(    ,
①𝑏< −1† ②0 < 𝑎<
1
2 †③𝑏+ 𝑎< 0†  ④𝑎−𝑏< 1.
Ak①②③
Bk②③④
Ck②④
Dk③④
7k(2024·ÐĂċó·Æö÷ø,yi0 < 𝑥< 𝑦< πIée𝑦sin𝑥= e𝑥sin𝑦IC e .Ę¿à
%ę
IãØŊ
űCW>56%B(    ,
Akcos𝑥+ cos𝑦< 0
Bkcos𝑥+ cos𝑦> 0
Ckcos𝑥> sin𝑦
Dksin𝑥> sin𝑦
8k(23-24 ‡Ø·ÐĂŲì·śŭŮů,yiŠ
𝑓(𝑥) = e𝑥+𝑎𝑥2¦¸31𝑥1,𝑥2Ié𝑥1 > 𝑥2IãØŊįv
7ÿ¹%B(    ,
Ak𝑎< −e
Bk𝑥1 + 𝑥2 > ln(𝑥1𝑥2) +2
Ck𝑥1𝑥2 > 1
Dk𝑓(𝑥)20ú'1
‡@ų
9k(2024·åĐåě·Æö÷ø,āŠ
𝑓(𝑥) = 𝑥ln𝑥+ (1−𝑥)ln(1−𝑥)Iã(    ,
Ak𝑓(𝑥) = 𝑓(1−𝑥)
BkŠ
𝑓(𝑥)2&c'−ln2
Ck¢𝑥1 + 𝑥2 = 1Iã𝑥1𝑓(𝑥2) + 𝑥2𝑓(𝑥1) ≥−ln2
Dk¢𝑥1 + 𝑥2 < 1Ié
1
2 < 𝑥2 < 1Iã𝑓(𝑥2) < 𝑓(𝑥1)
10k(2024·ċČċÛ·WÆ,yiĆ=𝑦= 𝑘𝑥/ç=𝑦= ln𝑥¨ĪÁ7¦1𝑀(𝑥1,𝑦1)I𝑁(𝑥2,𝑦2)Iç=𝑦= ln
𝑥M1𝑀è%<=/M1𝑁è%<=¨ĪÁ1𝑃(𝑥0,𝑦0)Iã(    ,
Ak0 < 𝑘<
1
𝑒
Bk𝑥1𝑥2 = e𝑥0
Ck𝑦1 + 𝑦2 = 1 + 𝑦0
Dk𝑦1𝑦2 < 1
11k(2023·ċČċÛ·WÆ,yi𝑎> 0，𝑏> 0，𝑎𝑏e𝑎+ ln𝑏−1 = 0Iã(    ,
Akln𝑏>
1
𝑎
Bke𝑎>
1
𝑏
Ck𝑎+ ln𝑏< 1
Dk𝑎𝑏< 1
‘@Ŵŵ
12k(2023·ìíŶŷ·Æö÷ø,yi𝑡> 0I𝑡𝑚2(ln𝑚−ln𝑛) +𝑡𝑛2 +𝑡𝑚𝑛−𝑡2𝑚𝑛≥𝑚𝑛456Iã𝑡=      
k
13k(2023·Ĉĉ·Æö÷ø,¢àÁ∀𝑚∈[−e,e]I∀𝑦∈(−1, + ∞)Iò˜7894𝑥3 + ln(𝑥+ 1) + (2023−𝑚)
𝑥−1 < 𝑦ln(𝑦+ 1)456Iã¬
 x %&c'.      
k
      
      
14k(2023·ûĐŸÛ·Æö÷ø,yiŠ
𝑓(𝑥) =
1
2𝑥2 + (1−𝑎)𝑥−𝑥ln𝑥2¦¸0'1𝑥1,𝑥2(𝑥1 < 𝑥2)Iãä
 a
%)'*+.      
†¢3𝑥1 ≥𝑥2Iãln𝑥1 + ln𝑥2 +2𝑎%&c'.      
k
Ð@Z[
15k(2024·ÙÚŹź·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑥2
e𝑎𝑥IC𝑎> 0k
(1)¢𝑓(𝑥)M(0,2]ß!"ŻĩIg𝑎%)'*+†
(2)ê𝑎= 1ëI¢𝑥1 + 𝑥2 = 4é0 < 𝑥1 < 2Iżb𝑓(𝑥1)/𝑓(𝑥2)%cúIƒį;x
16k(2024·ėĐŽž·Æö÷ø,yiŠ
𝑓(𝑥)%>?ſ.(0, + ∞)IAŠ
𝑓′(𝑥) = 2𝑥+
2
𝑥−2𝑎(𝑎∈𝑅),𝑓(1)
= 1−2𝑎k
(1)gç=𝑦= 𝑓(𝑥)M1(1,𝑓(1))è%<=𝑙%u·IƒŽ𝑙BĬƀ¥W¸>1†
(2)¢∃𝑥1,𝑥2I€0 < 𝑥1 < 𝑥2Ié𝑓′(𝑥1) = 𝑓′(𝑥2) = 0Ig2𝑓(𝑥1)−𝑓(𝑥2)%)'*+k
17k(2024·óôÏõ·Æö÷ø,yiŠ
𝑓(𝑥) = 𝑎(1−2ln𝑥) + 4𝑥6(𝑎∈R)k
(1)ðñ𝑓(𝑥)%!"#†
(2)¢𝑥1I𝑥2(𝑥1 ≠𝑥2).Š
𝑔(𝑥) = 𝑘𝑥2 +
1
𝑥2−ln𝑥%¦¸31Ig:
(𝑥1𝑥2)4 > 12e4k
18k(2024·ÐĂƁď·‡Æ,yiŠ
𝑓(𝑥) = ln𝑥+ 𝑥2−2𝑎𝑥,𝑎∈RI
(1)ê𝑎> 0ëIðñ𝑓(𝑥)%!"#†
(2)¢Š
𝑓(𝑥)2¦¸0'1𝑥1,𝑥2(𝑥1 < 𝑥2)Ig2𝑓(𝑥1)−𝑓(𝑥2)%&ú'.
      
      
19k(2024·Ĉĉ·Æö÷ø,yiŠ
𝑓(𝑥) = e𝑥−𝑎𝑥2
1+𝑥2 3 ¸0'1𝑥1,𝑥2,𝑥3ICeBĘ¿à
%ę
k
(1)gä
𝑎%)'*+†
(2)g:
𝑥1 + 𝑥2 + 𝑥3 > −2k