(Can you smash the recommendation ticket for me?)

Brush brush, brush, and soon Chen Jianyong wrote this finale inscription on the blackboard.

1. Known function f(x)=\frac{1+\ln(x+1)}{x}

(1) Find the defined domain of the function;

(2) determine the monotonicity of the function f(x) on the defined domain and prove your conclusion;

(3) When ...... frac{k}{x+1} is constant, and the maximum value of the positive integer k is found.

"You read that right, this is a finale related to functions. Chen Jianyong said lightly.

"Oh~ It's a function problem again, it's annoying. ”

"When I see it, I'm afraid. This kind of finale question is the most annoying!"

"It feels like it's so hard!"

"This finale question seems to be much more difficult than the usual finale question of the exam, is it carefully prepared for us by the teacher?"

The next group of classmates sighed and shook their heads.

"Okay, okay, don't complain, you will definitely get the college entrance examination for the inequality of function derivatives, conic curves, etc.!" So no matter how painful you are, you have to overcome them, okay?" Chen Jianyong said in a deep voice. I have to say that this finale question was indeed carefully prepared by him for these students below. This finale question was not made by him, but he cheekily asked his college math teacher to help him. Even it took him half an hour to solve this problem. It's definitely the finale of hellish difficulty!

Then there was a moment of silence in the classroom. Because all the students looked up at the questions on the blackboard and began to think hard.

Xiao Jun also read the question once, and then began to rack his brains and start thinking.

Soon, the first sub-problem, that is, the first problem of finding the definition domain of the function, he had the idea of solving the problem.

Originally, like this kind of finale question, often the first small question is a foundation, and many people can generally answer it as long as the foundation is good.

Xiao Jun was originally a student in a key class, and his previous foundation was still there, plus he read all the high school math books two days ago. The foundation is even more solid.

But after reading the second and third questions, he had no idea at all!

This is no longer a question of whether the foundation is basic or not, it has something to do with your thinking, the contact bypass after doing similar problems, and your IQ.

"Untie the hair and throw it to the top students in the group. Xiao Jun didn't plan to solve the problem by himself at all, so he directly took pictures of the questions on the blackboard with his eyes and threw them directly into the Xueba chat group.

【Xueba Chat Group】

Xiao Jun: @泷岛彗快点出来, help me solve this problem!

Takishima Comet: Good group leader~

"Hey, this kid will come out as soon as Aite, not bad. "Xiao Jun is very satisfied with Takishima's performance!

Time passes little by little.

Xiao Jun also looked at the time, he wanted to see how many minutes this omnipotent Takishima comet could solve this problem.

"Hey, a minute has passed!"

"Two minutes have passed!"

"It's been three minutes!"

At this moment, a message came from the group.

Sheldon Lee Cooper: @肖俊, leader, I've got it!

Answer:

Untie:

(1) The domain of f(x) is (-1,0)\cup (0,+\infty);

(2) f'(x)=\frac{\frac{x}{x+1}-\ln(x+1)-1}{x^2}，

So the positive or negative of f(x) is related to -\frac{1}{x+1}-\ln(x+1),

While............ So g(x) is monotonically increasing over (-1,0............ f(x) decreases monotonically on (-1,0).

(3) Method 1: Parametric variable separation

Namely:............ Let h(x)=(x+1)\frac{1+\ln(x+1)}{x},

then h'(x)=\frac{1}{x^2}(x-1-\ln(x+1)),

When x=a, h'(x)=0, i.e., a-1-\ln(a+1)=0 (*)

On either side of a, h(x) first decreases monotonically and then monotonically...... A\in (2,3) can be determined by the * formula,

So a+1\in (3,4),

So k_{\max}=3;

"Damn, this thing has been peeping at the screen?" Xiao Jun was very surprised that Sheldon Lee Cooper took the initiative to solve the problem for himself. Is it to avenge the loss of face in the past? Does he want to show in front of him that his solution speed is faster than that of Takishima Comet?

Takishima Comet: Leader, I've already solved it...... Hey...... Sheldon Lee Cooper You ......

When Takishima Comet saw the answer in the group, his face suddenly fell. Nima ...... Specifically against me?

Sheldon Lee Cooper: @泷岛彗, I just tried to solve it when I had nothing to do~

Takishima Comet: It's good if you're happy~

Now Sheldon Lee Cooper is so proud in his heart, because he solves the problem 5 seconds faster than Takishima Comet!hahahaha...... He felt that he was still no worse than Takishima Comet.

Looking at this 2B young man Sheldon Lee Cooper, he secretly said in his heart: "Sure enough, the 2B youth is much happier~"

But I have to say that Sheldon Lee Cooper is really a genius student, and such a super difficult finale question can be solved in three minutes, which is simply invincible. Well, Takishima Comet is not bad, a few seconds of time difference is equal to half a catty and eight taels for everyone.

Without saying anything, Xiao Jun immediately wrote this answer on the homework book.

Brush brush brush ~

It was written in a minute.

At this time, only five minutes had passed since Chen Jianyong's math teacher finished writing the problem.

I saw Xiao Jun take the meal card, take the dormitory key, pick up the homework book and stride to the podium.

On the other hand, the students in the classroom are all racking their brains and meditating, or they are grabbing their hair and looking at the blackboard in a daze, and some are biting the pen and can't put it down.

Even if Yang Qianqian was only on the first day of the class, she was now looking at the blackboard with a slight frown, probably thinking about the idea of solving the problem.

Xiao Jun stood out from the crowd and attracted the attention of the whole class, including math teacher Chen Jianyong.

"What is he doing up there?"

"Going to the bathroom?"

"It's not going to be solved, is it? It's only been five minutes!"

"Definitely not~ It's definitely going to the tuba, people have three urgency. ”

"I also think it's impossible to solve it in five minutes~"

"Teacher, I'm done. Xiao Jun said and put the homework book in front of the teacher.

"Karma?"

The whole class was shocked when they heard it one by one!

The mouths of each one are open, and they can stuff a duck egg!

"It was solved in five minutes?"

"Oh my God, isn't it!"

"How did he do it?"

"Is this the difference between a top student and a scumbag?"

"Ten times as many brain circuits as mine?"

Even Yang Qianqian was stunned when she saw this. Because she has not had an idea of the question of the second question until now, and she has already written out the answer to the first question. As for the third question, she hasn't read it yet.

But the other party solved it like this?

"No, no, no, the answer is definitely not right to solve it so quickly!" Yang Qianqian hinted to herself frantically in her heart.

Not only Yang Qianqian, but all the other students are thinking like this at the moment!

"It's been solved so quickly, I'll take a look!" Even Chen Jianyong was stunned. Although he knew that Xiao Jun had a full score in the monthly exam this time, it proved that he was strong. But this finale can't be done in five minutes, right?

(The question is not displayed normally, but there is no way, you can make do with it, anyway, you don't solve the problem.) ）