Xueba chat group

Chapter 21 3-Minute Problem Solving
(Can you give me a recommendation ticket?)

After a while, Chen Jianyong quickly wrote this finale inscription on the blackboard.

1. Known function f(x)=\frac{1+\ln(x+1)}{x}
(1) Find the domain of the function;
(2) Determine the monotonicity of the function f(x) on the domain of definition, and prove your conclusion;
(3) When...frac{k}{x+1} holds constant, find the maximum value of the positive integer k.

"You read it right, this is a final question related to functions." Chen Jianyong said flatly.

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

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

"It feels, it looks so difficult!"

"This final question looks much more difficult than the final question of the usual exam. Did the teacher carefully prepare it for us?"

The group of students below were all sighing and shaking their heads.

"Okay, okay, don't complain, you will definitely pass the college entrance examination of function derivative inequalities and conic sections! So no matter how painful you are, you have to overcome them, okay?" Chen Jianyong said in a deep voice.It has to be said that this final question was indeed carefully prepared by him for the students below.He did not come up with this final question, but he had the cheek to ask his university math teacher to help him out.Even for him, it took half an hour to solve this problem.It is definitely the finale question of hell-level difficulty!
After that, the classroom fell into a moment of silence.Because every student looked up at the topics on the blackboard and began to think hard.

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

Soon, the first sub-problem, that is, to find the domain of the function, he has the idea to solve the first problem.

Originally, for this kind of finale question, the first sub-question is often about the basics, and many people can usually answer it as long as they have a good foundation.

Xiao Jun was originally a student in the key class, and his previous foundation was still there, and he had read all the math books in high school two days ago.The foundation is 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 not basic. It has something to do with your thinking, your comprehension after doing similar topics, and your IQ.

"Excuse me, let's throw it to the bullies in the group." Xiao Jun didn't intend to solve the problem by himself at all, he just took pictures of the questions on the blackboard with his eyes and threw them into the chat group of bullies.

【Xueba chat group】

Xiao Jun: @段岛辉 Come out quickly and help me solve this problem!

Comet Takishima: Good group leader~
"Hey, this kid comes out as soon as he hits, not bad." Xiao Jun was very satisfied with Takishima Comet's performance!
Time passed 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, 1 minute has passed!"

"2 minutes have passed!"

"3 minutes!"

At this moment, a message came from the group.

Sheldon Lee Cooper: @肖俊, group owner, I have solved it!
answer:

solution:
(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 sign of f(x) is related to -\frac{1}{x+1}-\ln(x+1),

While ... ... so g(x) is monotonically increasing at (-1, 0) ... f(x) is monotonically decreasing at (-1, 0).

(3) Method [-]: Separation of parameters
That is:...... 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, that is, a-1-\ln(a+1)=0 (*Formula)
On both sides of a, h(x) first monotonically decreases and then monotonically increases...A\in (2, 3) can be determined by * formula,

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

So k_{\max}=3;

"Fuck, this guy has been spying on 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?Does he want to show in front of him that he can solve problems faster than Comet Takishima?
Comet Takishima: Group lord, I have already solved it... Huh... Sheldon Lee Cooper, you...

When Comet Takishima saw the answer in the group, his face immediately collapsed.Nima... specifically against me?
Sheldon Lee Cooper: @段岛辉, I just tried to solve it when I had nothing to do~
Comet Takishima: Just as long as you are happy~
Now Sheldon Lee Cooper is so proud, because he can solve the problem 5 seconds faster than Comet Takishima!Hahahaha...he felt that he was no worse than Comet Takishima.

Looking at this 2B young man, Sheldon Lee Cooper, he thought to himself: "Sure enough, 2B young people are so much fun~"

But I have to say that Sheldon Lee Cooper is really a genius. He can solve such a super-difficult finale question in 3 minutes, which is simply invincible.Well, Comet Takishima is not too bad, a few seconds behind is equal to everyone.

Without further ado, Xiao Jun immediately wrote the answer in his homework book.

brush brush~
It was written in 1 minute.

At this time, only 5 minutes had passed since Chen Jianyong's math teacher finished writing the topic.

Xiao Jun took the meal card, the key to the dormitory, picked up his homework, stood up and strode towards the podium.

On the other hand, those students in the classroom are currently racking their brains and thinking hard, or they are holding their hair and staring at the blackboard in a daze, and some are biting their pens and cannot write.

Even Yang Qianqian, who is the first genius in the class, is looking at the blackboard with a slight frown, probably thinking about how to solve the problem.

Xiao Jun stepped onto the podium so out of the crowd, which attracted the attention of the whole class, including the math teacher Chen Jianyong.

"What is he doing up there?"

"To the restroom?"

"Could it be solved? It's only been 5 minutes!"

"Definitely not~ Definitely going to the tuba, people have three urgencies."

"I also think that it is impossible to solve it in 5 minutes~"

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

"Ka?"

Everyone in the class was instantly shocked after hearing this!
The mouths of each one are wide open, and a duck egg can be stuffed in!
"Solve it in 5 minutes?"

"My God, no!"

"How did he do that?"

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

"The brain circuits are ten times mine?"

Even Yang Qianqian was shocked when she saw this.Because until now, she has no idea for the second question, and she has already written 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, it's definitely wrong to solve it so quickly!" Yang Qianqian hinted frantically to herself.

Not only Yang Qianqian, but all the other students are thinking this way at the moment!
"It was solved so quickly? Let me see!" Even Chen Jianyong was stunned.Although he knew that Xiao Jun got a full score in the monthly exam this time, which proved that he is capable.But it's impossible to solve this final question in 5 minutes, right?
(The question display is not normal, but there is no way, you can make do with it, anyway, you can't solve the question.)
(End of this chapter)