已知函数f（x）=mx3+nx2（m，n∈R，m＞n且m≠0）的图象在（2，f（_爱下问搜题

搜题请搜索小程序“爱下问搜题”

问题解答题

已知函数f（x）=mx³+nx²（m，n∈R，m＞n且m≠0）的图象在（2，f（2））处的切线与x轴平行．

（1）求m与n的关系式及f（x）的极大值；

（2）若函数y=f（x）在区间[n，m]上有最大值为m-n²，试求m的值．

答案

（1）∵f′（x）=3mx²+2nx

由图象在（2，f（2））处的切线与x轴平行，知f′（2）=0

∴n=-3m，m＞0 ①

令f′（x）=3mx²+2nx=3mx²-6mx=0

得x=0或x=2，

∴f（x）在（-∞，0）上是增函数，在（0，2）上是减函数，在（2，+∞）上是增函数

∴x=0是f（x）的极大值点，x=2是极小值点．

∴极大值为f（0）=0；

（2）令f（x）=f（0）=0，得x=0或x=3

（I）当0＜m≤3时，f（x）_max=f（0）=0，∴m-n²=0 ②

由①，②解得m=

1

9

，符合前提0＜m≤3．

（II）当m＞3时，f（x）_max=f（m）=m⁴+m²n

∴m⁴+m²n=m-n² ③

由①，③得m³-3m²+9m-1=0，

∵m＞3时，m³-3m²+9m-1=m²（m-3）+9m-1＞0

∴m³-3m²+9m-1=0在（3，+∞）上无实数根．

综上讨论可知，m的值为m=

1

9

不定项选择

高锰酸钾试液鉴别可巴比妥类药物时( )

A．一般在酸性溶液中进行

B．一般在中性溶液中进行

C．一般在碱性溶液中进行

D．在加热条件下进行

E．在冷却条件下进行

完形填空

This morning, I got a funny feeling. Today I had to give a report in my        , and I had been afraid to do so for a long time.
Mom came into the kitchen, asking , “Are you ready,         ?”
“ Yes.” I answered quietly.         in fact I wasn’t ready!
On my way          school, I could feel my heart beating. I hated this!
I went into Mrs. Owens’ class. She was smiling, and saying        to all her students as she did every morning.
“ Hi, Sam,” she said to me as I walked         the classroom.
“ Hi,” I said and quickly sat in my        . I looked at the clock. The time was near. Then Mrs. Owens started, “ Sam, it’s time for you to show yourself.” “Show         ? Oh, no!”
I got up slowly,          the ground as I walked to the front of the class. Then, I looked up, and everyone was looking at me. Mrs. Owens knew I was         , so she came beside me, and put her hand on my back.
She said to the class, “Sam has worked very hard on his         ,and I know it is a very interesting report.” Then she turned to me and asked, “Sam,          did you like best about your report on Hawaii?”
I thought about the fun things I         about surfing before, and I started to talk. I talked and talked        the whole class, and I was not nervous at all. It was great fun. It wasn’t frightening. The class cheered as I finished and Mrs. Owens was         .
In fact, I can do better than I think I can— if I try! I am proud of myself.

C．dinning room

D．reading room

A．looking after

D．looking for

A．in front of