问题 解答题

(12分)

已知x=1是函数f(x)=mx3-3(m+1)x2+nx+1的一个极值点,其中m,n∈R.

(1)求m与n的关系式;

(2)求f(x)的单调区间;

(3)当x∈[-1,1]时,m<0,函数y=f(x)的图象上任意一点的切线斜率恒大于3m,求m的取值范围.

答案

(1)

(2)当时,单调递减,在单调递增,在上单调递减.

当m>0时,f(x)在(1+)及(-,1)上单调递增;在(1,1+)上单调递减 .

(3)的取值范围为

近几年新课标高考对于函数与导数这一综合问题的命制,一般以有理函数与半超越(指数、对数)函数的组合复合且含有参量的函数为背景载体,解题时要注意对数式对函数定义域的隐蔽,这类问题重点考查函数单调性、导数运算、不等式方程的求解等基本知识,注重数学思想(分类与整合、数与形的结合)方法(分析法、综合法、反证法)的运用.把数学运算的“力量”与数学思维的“技巧”完美结合

解:(I)因为是函数的一个极值点,所以,即,所以

(II)当m=0时,上为增函数,在(6,+)上为减函数

当m≠0时,=

时,有,当变化时,的变化如下表:

1
00
      
调调递减极小值单调递增极大值单调递减
故由上表知,当时,单调递减,在单调递增,在上单调递减.

当m>0时,f(x)在(1+)及(-,1)上单调递增;在(1,1+)上单调递减 .

(III)由已知得,即

所以

,其函数开口向上,由题意知①式恒成立,

所以解之得所以

的取值范围为

阅读理解

Science Daily—Kids may roll their eyes when their mothers asks them about their school day, but answering her may actually help them learn. New research from Vanderbilt University reveals that children learn the solution to a problem best when they explain it to their mom.

“We knew that children learn well with their moms or with a peer, but we did not know if that was because they were getting feedback and help,” Bethany Rittle-Johnson, the study’s lead author and assistant professor of psychology at Vanderbilt’s Peabody College of education and human development, said. “In this study, we just had the children’s mothers listen, without providing any assistance. We’ve found that by simply listening, a mother helps her child learn.”

Rittle-Johnson believes the new finding can help parents better assist their children with their schoolwork, even when they are not sure of the answer themselves. Although the researchers used children and their mothers in the study, they believe the same results will hold true whether the person is the child’s father, grandparent, or other familiar persons.

“The basic idea is that it is really effective to try to get kids to explain things themselves instead of just telling them the answer,” she said. “Explaining their reasoning, to a parent or perhaps to other people they know, will help them understand the problem and apply what they have learned to other situations. We saw that this simple act of listening by mom made a difference in the quality of the child’s explanations and how well they could solve more difficult problems later on.

小题1:According to the passage, a mom had better ________.

A.give a kid some help when he is explaining his problem

B.just listen while a kid is explaining his problem

C.work together with a kid toward the solution to a problem

D.leave a kid alone when he meets a problem at school小题2: If she wants to help a child with his schoolwork, a mom ________.

A.should know the answer to a problem first

B.should pay attention to the child’s feedback.

C.may not know the answer herself

D.should not interrupt the child小题3:Who will be the least help to a kid when he is explaining, according to Rittle-Johnson?

A.The kid’s mom.

B.The kid’s grandmother.

C.A peer.

D.A relative the kid doesn’t know.小题4:Which of the following best gives the main idea of the passage?

A.Children learn better when their mom is hearing their explaining.

B.Children learn better if they find the solution to a problem themselves.

C.Parents should not provide any assistance to their children’s schoolwork.

D.A mom should listen more to their children when they have problems at school.

填空题