第二章-练习-P49

练习

1. 求下列函数的极大点与极小点. (1) y?2x2?x?1

17解: 2x2?x?1?2(x?)2?

4817 极小值点: (?,), 无极大值点.

48

(2) y?x?x2?2

19解: x?x2?2??(x?)2?

2419极大值点: (,), 无极小值点.

24(3) y?解:

1 2x?x?311 ?2111x?x?3(x?)2?2414极大值点: (?,), 无极小值点.

211

(4) y?|x|?|x?1|

?1x?1?解: |x|?|x?1|??2x?10?x?1

??1x?0?极大值点: (1,1), 极小值点: (0,?1). (5) y?|x2?5x?6|

?x2?5x?6x?3??2解: |x?5x?6|???(x2?5x?6)2?x?3]

?2??x?5x?6x?251极大值点: (,), 极小值点: (2,0),(3,0).

24

(6) y?|log2|x||

?log2xx?1??logx0?x?1?2解: |log2|x||??

??log2?x?1?x?0??log2?xx??1由函数图像知

极小值点: (?1,0),(1,0), 无极大值点. (7) y?|解:

12x?|x|?3| 4?12|(x?2)?4|x?0?12?4|x?|x|?3|?? 4?|1(x?2)2?4|x?0??4极大值点: (2,4),(?2,4), 极小值点: (0,3),(?6,0),(6,0).

(8) y?|x2?3x?2|?2?x2?3x

1?x?2?0?解: |x2?3x?2|?2?x2?3x??2(x2?3x?2)x?2

?2?2(x?3x?2)x?1极小值点: (1,0),(2,0). (9) y?2sinx?1

解: 由三角函数的性质得

?极大值点: (2k??,3),, 极小值点: (k?,1), k?Z.

2(10) y?sinx?3cosx

解: sinx?3cosx?10sin(x??),??arcsin极大值点: (2k??

(11) y?sin2x?sinx

3 103???,?10),, k?Z. 2?2??,10),, 极小值点: (2k??解:

?0sin2x?sinx????2sinx2k??x?(2k?1)?k?Z,

(2k?1)??x?(2k?2)?极大值点: (2k??(12) y?2解: 2x4?x2x4?x23?,2), 极小值点: (k?,0),k?Z 2

11(x2?)??2241?1极大值点: (0,1),, 极小值点: (?,24).

2(13) y?2sinx

解: 由三角函数性质得

??1极大值点: (2k??,2),, 极小值点: (2k??,), k?Z.

222(14) y?1?sin2x 解: 由三角函数性质得 ??极大值点: (k??,2),, 极小值点: (k??,0), k?Z

44(15) y?x4?x2?1

13解: x4?x2?1?(x2?)2?

24极大值点:(0,1), 极小值点: (?|x| 21?x23,). 24(16) y??1?1??x|x|?x解: ??11?x2???1?x??xx?0

x?01极大值点: (?1,), 极小值点: (0,0).

24?x| (17) y?|x?1解: 因为y?0, 所以极小值点: (4,0).

(18) y?tan|x|

?tanxx?0解: tan|x|??,

??tanxx?0由三角函数图像得极小值点: (0,0). (19) y?2e 21?x?x2e?x1解: ?22x21?xe(1?x)函数有极大值点: (0,1) (20) y?x 2(1?x)|lnx|解: 由函数性质得函数无极值点. (21) y?3|x?1|?|x?1|

?3?2??|x?1|?|x?1|??3?2x解: 3?2??3x?1?1?x?1 x??11极大值点:(?1,9), 极小值点:(1,).

9(22) y?(x?3)2(x?2)2

51解: (x?3)2(x?2)2?((x?)2?)2

2451极大值点: (,),极小值点: (3,0),(2,0).

216

(23) y?x2?x,x?[a,2]

11解: x2?x?(x?)2?

24由函数性质得 1当?a?2时, 函数无极值点. 2111当a?时, 有极小值点(,?).

224(24) y?解:

sinx

1?cosxsinxsinx ?x1?cosx2|cos|2函数无极值点.

2. 求下列函数的最大值与最小值. (1) y?x2?2x?3,x?[?2,4]

解: x2?2x?3?(x?1)2?2,

y

max[?2,4]?27,x?4,ymin[?2,4]?2,x??1,

(2) y?x2?x?1,x?[a,b]

13解: x2?x?1?(x?)2?

241当 a??,ymax?b2?b?1,x?b,ymin?a2?a?1,x?ax

2[a,b][a,b]1当 b??,ymax?a2?a?1,x?a,ymin?b2?b?1,x?b

2[a,b][a,b]11131当 a???b且??a?b?,ymax?a2?a?1,x?a,ymin?,x??

222[a,b]42[a,b]11131当 a???b且??a?b?,ymax?b2?b?1,x?b,ymin?,x??

222[a,b]42[a,b](3) y?11x?x?12

1 13(x?)2?24解:

x2?x?1?ymax?2R31,x??. 32x2(4) y?4

x?1

本文来源：https://www.bwwdw.com/article/1awf.html