From 47b7c5493caef04e8a3b51cd4b24575c68062882 Mon Sep 17 00:00:00 2001 From: xianliticn Date: Wed, 4 Feb 2026 13:18:58 +0800 Subject: [PATCH] =?UTF-8?q?Update=20=E6=96=87=E7=AB=A0=20=E2=80=9C?= =?UTF-8?q?=E5=A4=A7=E5=81=9A=E4=B8=80=E9=81=93=E5=B0=8F=E5=AF=BC=E6=95=B0?= =?UTF-8?q?=E9=A2=98=E2=80=9D?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- content/blog/大做一道小导数题.md | 28 ++++++++--------------- static/uploads/geogebra-export-2.png | Bin 0 -> 65143 bytes static/uploads/geogebra-export-3.png | Bin 0 -> 63973 bytes static/uploads/geogebra-export-4.png | Bin 0 -> 116207 bytes 4 files changed, 10 insertions(+), 18 deletions(-) create mode 100644 static/uploads/geogebra-export-2.png create mode 100644 static/uploads/geogebra-export-3.png create mode 100644 static/uploads/geogebra-export-4.png diff --git a/content/blog/大做一道小导数题.md b/content/blog/大做一道小导数题.md index 0390901..5fd5f64 100644 --- a/content/blog/大做一道小导数题.md +++ b/content/blog/大做一道小导数题.md @@ -3,9 +3,9 @@ title: 大做一道小导数题 date: 2024-01-03T23:56:00.000+08:00 author: 线粒体 --- - ## 题目 -**【2024·嘉峪关市酒钢三中1月联考预测卷:21. (2)】** $g\left ( x \right ) = x\ln_{}{\left ( x+1 \right ) } +a\ln_{}{\left ( x+1 \right ) } -x$ ,$g\left ( x \right )$ 在 $\left ( 0,+ \infty \right )$ 单调递增,求 $a$ 的取值范围. + +**【2024·嘉峪关市酒钢三中1月联考预测卷:21. (2)】** $g\left ( x \right ) = x\ln*{}{\left ( x+1 \right ) } +a\ln*{}{\left ( x+1 \right ) } -x$ ,$g\left ( x \right )$ 在 $\left ( 0,+ \infty \right )$ 单调递增,求 $a$ 的取值范围. ## 大做法 @@ -33,24 +33,18 @@ ${g}'' (x)$ 是 ${g}' (x)$ 的导函数,考察 ${g}'' (x)$ 来观察 ${g}' (x) 由上总结 ${g}' (x)$ 的两种单调性: -1. 当 $a-2>-1$ 即 $a>1$ 时,${g}' (x)$ 在 $a-2$ 左侧减,右侧增(图1). -2. 当 $a-2 \le -1$ 即 $a \le 1$ 时,${g}' (x)$ 在定义域上增(图2). +1. 当 $a-2>-1$ 即 $a>1$ 时,${g}' (x)$ 在 $a-2$ 左侧减,右侧增(图1). +2. 当 $a-2 \le -1$ 即 $a \le 1$ 时,${g}' (x)$ 在定义域上增(图2). - -
-
图1
- +![](/uploads/geogebra-export-3.png "图1") - -
图2
-
- +![](/uploads/geogebra-export-2.png "图2") 注意到我们要求的是命题 $p : \forall x\in (0,+\infty)$ ,${g}' (x) \ge 0$ 为真时,$a$ 的取值范围. 显然,对于 $a-2 \le -1$ 即 $a \le 1$ 的情况,由于 ${g}' (x)$ 在定义域上增,故在 $(0,+\infty)$ 上也增,因此只需使 ${g}' (0) \ge 0$ ,就可以使命题 $p$ 为真.所以 ${g}' (0)=a-1 \ge 0$ ,$a \ge 1$ .结合 $a \le 1$ 的前提,得 $a=1$ . -$a-2>-1$ 即 $a>1$ 的情况比较复杂,可以分为两种情况去讨论.**第一**,当 $-10$ 即 $a>2$ 时,${g}' (x)$ 在 $x=a-2$ 处取得最小值,故只需 ${g}' (a-2) \ge 0$ 即可满足命题 $p$ ,${g}' (a-2)=\ln_{}{(a-1)}+\dfrac {a-1}{a-1}=\ln_{}{(a-1)}+1 \ge 0$ ,$\ln_{}{(a-1)} \ge -1$ , $a \ge \dfrac {1}{\mathrm{e}} +1$ ,结合前提得 $a>2$ . +$a-2>-1$ 即 $a>1$ 的情况比较复杂,可以分为两种情况去讨论.**第一**,当 $-10$ 即 $a>2$ 时,${g}' (x)$ 在 $x=a-2$ 处取得最小值,故只需 ${g}' (a-2) \ge 0$ 即可满足命题 $p$ ,${g}' (a-2)=\ln*{}{(a-1)}+\dfrac {a-1}{a-1}=\ln*{}{(a-1)}+1 \ge 0$ ,$\ln_{}{(a-1)} \ge -1$ , $a \ge \dfrac {1}{\mathrm{e}} +1$ ,结合前提得 $a>2$ . 综上,列出满足 $p$ 的三个条件: @@ -74,14 +68,12 @@ $a-2>-1$ 即 $a>1$ 的情况比较复杂,可以分为两种情况去讨论.* 对于 $x\in (0,+\infty)$ ,$\ln_{}{(x+1)}$ 是恒正的.$\dfrac{a-1}{x+1}$ 则有两种情况:第一,当 $a-1 <0$ 即 $a<1$ ,$\dfrac{a-1}{x+1}$ 恒为负.第二,当 $a-1 \ge 0$ 即 $a \ge 1$ ,$\dfrac{a-1}{x+1}$ 恒非负. -如图3,对于 $x\in (0,+\infty)$ ,当 $a-1 <0$ 时,其他点暂且勿论,单看 $\lim_{x \to 0^{+}} \ln_{}{(x+1)}+\dfrac{a-1}{x+1} < 0$ ,就知道 $a-1 <0$ 不满足这个命题.而 $a-1 \ge 0$ 显然是满足的,因为两正数相加结果一定为正数.故只需使 $a-1 \ge 0$ 即可使命题为真. +如图3,对于 $x\in (0,+\infty)$ ,当 $a-1 <0$ 时,其他点暂且勿论,单看 $\lim*{x \to 0^{+}} \ln*{}{(x+1)}+\dfrac{a-1}{x+1} < 0$ ,就知道 $a-1 <0$ 不满足这个命题.而 $a-1 \ge 0$ 显然是满足的,因为两正数相加结果一定为正数.故只需使 $a-1 \ge 0$ 即可使命题为真. - -
图3
- +![](/uploads/geogebra-export-4.png) 该做法相当巧妙简洁,在考场上能够节省大量时间,是绝大多数学生采用的做法. > 题不要多做,要做精、做透,研究得清清楚楚。 -> +> > ——@nanmu diff --git a/static/uploads/geogebra-export-2.png b/static/uploads/geogebra-export-2.png new file mode 100644 index 0000000000000000000000000000000000000000..d00efb95eb5648836bc6e2676175215e0010b1e8 GIT binary patch literal 65143 zcmeFacTiJV*f6dlqN3p1P^zv4L5dZSx`K*QWCaAJihxKD5Qq>GR7%!GM5T$8EFvJ% zdk7^!l-@fep@`H#A|(lMN%D*9S~k}|zVDsi%sb!A&al^;d(J)QY3FI@Id>oZepP$> zR^hGd)~(xq(=qD-^{~#Ql2gQkn_(*hx7X9*R9J7=U=no<@{dnWvH#W zF1uB9n)71|lt7C^y$-)BS)xIYI}S8($W$D0ORB1 zY&N^9s%m+8d2w-Zetv#rWaLJS`ReNG+S=N}!UBWApwVb6D=SPUlTN3zSS&J`JT)~X zBO^06H+Mbq28BX#b#=|p&u?yS{`m3Z?CflRfB)p<#Kgq(^t4IzO*zCVvlo`S ze_pkUvuSN@ZEI_DPIkMUU_Uf7bnMtM|LlO(wbh>S-U}BlKvKQ#W%#A1r;|verRAmK z;^Mn^?>;IDo%}SJ(U|oN8v%XeSJ!j6Gsjsh}r_aY##y`jo%FfOX z4GpzN-+lM)9hLS;Sy}nF-+t54(Yb&B{u+}BtO4cap_5OYA z!C}31>kh2Da^bx3y}?Ms#_ z9!z~BuDskfqxf1?3!<)Ef#9&WSW(-7qV^IdzHNliGMPqVVdMGwuBWS}%ZUhAX3D{@ zHOkFU7XSyqd8Z7ut`l<28fG%nn$aXza4tSFzrfC4BQk6y2|KImf;r|b|F+30I)AcZ zRWAQr5MiMl8rM5)FGYSnp71ypAHI00&wJV@%2*%}=cbIhh=*R`(`ZRVk!7H5`1XW0 zNqgJMHp5Dfsl=n?xI6N=vCX74SGZ`Bh#=BF|Fp$t`v`{C$DLf z6YURFFoFLkfAC!x4Y(SbZG}^Z@WKA|=f0WY5FVZrz^0VU@IL%D7$6h`@1qqkY3;my z`Nr9An?gd$X7atI2))DO`myt_x#d?Jl`?Y5?JKl95(ya+%{(L;Ew%Nw;=2A%=HH2K^x{xa$LE9ck#nH~an|`ifai>jd ztbGnk$Umug6JMJ|kRO)a%m7h}fm=?@Hw?bslsp9ZN=%zLUH<(iQq8n2>1+J=Lrxz;71OB1^vCD8 z5z%<*G2{JdjY`)iye1pAS+MK0J#%iP8rgq)UYmaEYShZZw8r~?&&yUUVb8x^TWK2| z4&UR+MRGpICP)BXsfroSk{{2(Ri-m11Sli_{zIXFE(RUR5}+(9Mh$brUd}`+ zRyR_jZf*RxC~}kOzt;aK@c$qMavlN=4bUQ7>(z(kSH@N*F~Qq&bO&CuE!g2K4MkK185m&sO+9qJ zDQ((guZ6et9>7MrlBo-+&C3cmhF+hE@987O^=a+T2XR(&R`T$Jg-^qRVY7ZU$$B%id>-vjweKgG_7J$hCbIUiyMpW zuQiHuSYf6x(dQy?9zXtv>-2NlTs{XYXo|Unn{Iyj`xocpug6pb!o3GxLpkP6WRu29 z@8pg&-ND?pMAQ{Rd%{&J%G9|x*JCO>f<_x?qAF=BO00PVxv4H+OCp@I^tsvS{0S|wz+rdp!ib`g{tI#b?%MNE5YB2eQ=dl;XFG}Jt`NkMnTzUybs?Y|v7 z_WM?`r87|%*RP|MFk+=4BVXF~{?C z-#yWOD@FN!`8Gm>W5#>O!2Z0Wb=6sv!W3zkHHgZbJ5oJw9(nq%4D_{#FWzebiCpXr?J~;~yjd<#ZJR!GLI&3u-y7=U?h&m_ zJbpDJwK0BAn)jm9Ts`Y;4a_+U|E89*Q+Dlyj|S9MA0|t*^jWB$t;71a=)=zD<8Fz4 z<_WrZu>*<^MyF5r4p?r=lN(MXLCtf~nO)4e)J2{6`F(m{Q^R9kp zzE|ke+Ml%!^FyJ{RdS^D_?_%|lUwi(UYOU=UrMg>Ca7yy31N}$DiX~;9)kxkwc&S#VJ6aD|BdtB9R$LYwx=NTC>`@fE zgOPamBt?QES7Ao}-2wJ!FH#ksw`8jbmn^LCKN08E0fqevh}4NAIl5=e`vphg%B|_? 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