1 . 已知函数
的定义域为D,若对任意的实数
,都有
成立(等号当且仅当
时成立),则称函数
是D上的凸函数,并且凸函数具有以下性质:对任意的实数
,都有
(
,
)成立(等号当且仅当
时成立).
(1)判断函数
、
是否为凸函数,并证明你的结论;
(2)若函数
是定义域为R的奇函数,证明:
不是R上的凸函数;
(3)求证:函数
是
上的凸函数,并求
的最大值(其中A、B、C是
的三个内角).
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dbc1bc250c8a6523a1be394ff48d4a51.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39243f2c10a8291d75d65694b2dec94a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7f333263260646c494225db8a7476c00.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/942c2141d01bde6b48210c56a17fc75e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3bd2179ba09ac27fce32baf170528ea6.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ddec2919fb0760a9b54440e581d6f7a4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ac69e6db1df13ed64756b4f391ae9fac.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e167b43045b3297248e334c41c621b8f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/73223617c8855826298d435673787a94.png)
(1)判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c57e815c01a412466a6aa12d3e883a77.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/128d3060b444b2ee0ef61f2420c5109b.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1938c093dd2fbcb752d0eb7a18d143b2.png)
(3)求证:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/d2b9643da0c0fea4f099f9a9133d6076.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/71163f419555f2ed76075c8ff659fbfc.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a8080fef9bdfa92ae70f3e314eef3e3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
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2023高一上·全国·专题练习
2 . 求证:
(1)
;
(2)
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/4efaf5d28f30a4a834d74506fe633767.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/50cca9624f8dbeb8f6a5e315e09d0846.png)
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2023高三·全国·专题练习
3 . 在
内存在一点
,满足
,求证:
的三边构成等比数列.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1dde8112e8eb968fd042418dd632759e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e333acaac72dde213004917608ae340c.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e428e7a09732be85c1224e9c8f6a71c5.png)
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2023高三·全国·专题练习
解题方法
4 . 已知
,
,求证:
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/694778ef800750095f750d1ca798814e.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dcae870077ceb017dac3423b7b53c7ad.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5479187a07b26f83eb12ea30e1e7ab16.png)
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5 . 求证:
(1)
;
(2)
.
(1)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/5a431fc00c5aadc9162678c49e041316.png)
(2)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c4e3ab9ca6742361e8681ed42e36e2f0.png)
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6 . 记
的内角
的对边分别为
,已知
.
(1)证明:
;
(2)若
的面积为
,求
的最大值.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/24e0c10fb103930eabd5fa18e8f9bb06.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/76f0649064a085fb74c997fb507a9b6d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b31001c9f106ec7c293fbc433bbc597e.png)
(1)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/02746ec8e4220d8b4a174d5e9a711ed2.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/15c0dbe3c080c4c4636c64803e5c1f76.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cf231f8f86fb922df4ca0c87f044cec3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/86279407f407a070f7925563127305e9.png)
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7 . 设
是定义域为
的函数,如果对任意的
、
均成立, 则称
是“平缓函数”.
(1)若
, 试判断
和
是否为“平缓函数” ? 并说明理由; (参考公式:
时,
恒成立)
(2)若函数
是“平缓函数”, 且
是以 1为周期的周期函数, 证明:对任意的
、
, 均有
;
(3)设
为定义在
上函数, 且存在正常数
使得函数
为“平缓函数”. 现定义数列
满足:
, 试证明:对任意的正整数
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/1076aeb555debc8bf15c586375d64c83.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
(1)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc2fe919afe3583b1ef0dadbd23102a0.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/70458945dae62f8f6a553dfaa8eb723a.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/423b40b099989abb2d15ddf60413c8a2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/08115d6d9f876dea921a4d32260ff1fb.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/640461197eaf70c0ebe1eb89daf383c1.png)
(2)若函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/51c530f4b7491b95acb8ce3eef9aa09d.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/c814128ea2139e33db94ea590e7c2223.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/33890c6b0bf167514d44139d9dca0154.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/49a2b43fdce5aaae58c0907de23cbc6c.png)
(3)设
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a43b2faa4f81f32d94612dce724e772b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ee6bf90a1bbeea09e1b7206975a99f5f.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b66efdaa9fe939b2b1a7326aaf5e35d3.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e976c0663fa749ca749f99842d21ca03.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/62f422484ad3a17eeebc4d15a3fb6289.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/848ea90ecbd6c506a78058f983ff5614.png)
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名校
解题方法
8 . (1)证明:
;
(2)若
,
,其中实数
,
不全为零.
①求
;
②求
.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fede4c333b02961cce4709facfa6893f.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/756e54aea128da691443988955aa41b4.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/43e68ff9b088253c2a462a7ac7f9d8cf.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0a6936d370d6a238a608ca56f87198de.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2c94bb12cee76221e13f9ef955b0aab1.png)
①求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/bbb006ea697b63a914eb487073f0abe1.png)
②求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/cc3884b343d76a26b4b85b48987d7064.png)
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9 . 设锐角三角形ABC的内角A、B、C所对的边分别为a、b、c,已知
.
(1)求证:B=2A;
(2)求
的取值范围.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6194c583037bd04e32fbf9b435084a9d.png)
(1)求证:B=2A;
(2)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/fd950ec83d93596468e3aff0bb91e0e9.png)
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2022-12-29更新
|
5082次组卷
|
7卷引用:专题4 三角函数与解三角形 第2讲三角恒等变换与解三角形
(已下线)专题4 三角函数与解三角形 第2讲三角恒等变换与解三角形安徽省阜阳市第四中学2023届高三下学期第一次月考数学试题第11章《解三角形》单元达标高分突破必刷卷(培优版)陕西省渭南市韩城市象山中学2022-2023学年高一下学期期中数学试题广东省汕头市2023届高三上学期期末数学试题(已下线)高一下学期期中复习解答题压轴题十八大题型专练(1)-举一反三系列(人教A版2019必修第二册)(已下线)专题02 高一下期末真题精选(1)-期末考点大串讲(人教A版2019必修第二册)
名校
解题方法
10 . 令
,
,定义函数
,如果
,则称非负整数n为好整数,所有好整数的集合记作W.
(1)求
、
的值;
(2)证明:
;
(3)求出集合W.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/721c92cf1759acf10d2e74f6e4915158.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/39f7a420205bbe7fb7a5707a14fd3a1b.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/22d71c5943b3d883e8721a4c5bbee5e1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a6b91e45df2d12396d9dbdf8748fec07.png)
(1)求
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/3e38fffbc7ab9882480f4faa72390e23.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/7ce6155e181e21ce56ea658b70f8af17.png)
(2)证明:
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/a5398b3972437e94931fbbc9504f80d3.png)
(3)求出集合W.
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