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极值拐点学习笔记

  • 2026-03-24 05:07:33
极值拐点学习笔记

背景：

在学习极值拐点的时候，总是不停的去背“三充分一必要”条件，可是在做题时候，却一直不是很顺畅，对于很多人来说容易的定义，对于我总有那么点说不清道不明的点卡着，这次通过一道不等式题，才让我对其条件有了更进一步的理解。特别是第三充分条件
第三充分条件定义：
极值：
拐点：
从定义可以看出，极值是从f'(x)=0开始，拐点是f''(x)=0，之后在求到n阶导，根据n为奇/偶来判定，以前以为关键点是后续的n,其实还有个关键点就是他们是由谁开始的，极值从f'一阶导开始，正如他的必要条件只要是极值一定满足f'(x)=0. 拐点亦是。
所以在运用第三充分条件的时候，首先确定是极值还是拐点，优先看一阶导，在依次看后续的n阶来判定才有意义
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  1. CONNECT:[ UseTime:0.000483s ] mysql:host=127.0.0.1;port=3306;dbname=no_67808;charset=utf8mb4
  2. SHOW FULL COLUMNS FROM `fenlei` [ RunTime:0.000817s ]
  3. SELECT * FROM `fenlei` WHERE `fid` = 0 [ RunTime:0.008986s ]
  4. SELECT * FROM `fenlei` WHERE `fid` = 63 [ RunTime:0.001366s ]
  5. SHOW FULL COLUMNS FROM `set` [ RunTime:0.000638s ]
  6. SELECT * FROM `set` [ RunTime:0.005712s ]
  7. SHOW FULL COLUMNS FROM `article` [ RunTime:0.000673s ]
  8. SELECT * FROM `article` WHERE `id` = 476736 LIMIT 1 [ RunTime:0.000464s ]
  9. UPDATE `article` SET `lasttime` = 1774347267 WHERE `id` = 476736 [ RunTime:0.016873s ]
  10. SELECT * FROM `fenlei` WHERE `id` = 65 LIMIT 1 [ RunTime:0.007219s ]
  11. SELECT * FROM `article` WHERE `id` < 476736 ORDER BY `id` DESC LIMIT 1 [ RunTime:0.005870s ]
  12. SELECT * FROM `article` WHERE `id` > 476736 ORDER BY `id` ASC LIMIT 1 [ RunTime:0.007560s ]
  13. SELECT * FROM `article` WHERE `id` < 476736 ORDER BY `id` DESC LIMIT 10 [ RunTime:0.023872s ]
  14. SELECT * FROM `article` WHERE `id` < 476736 ORDER BY `id` DESC LIMIT 10,10 [ RunTime:0.003213s ]
  15. SELECT * FROM `article` WHERE `id` < 476736 ORDER BY `id` DESC LIMIT 20,10 [ RunTime:0.001623s ]
0.156400s