On the first day of the Lunar New Year, Chen Zhou spent this kind of efficient problem-solving.

There are still 4 and a half cans of psychotropic potions left.

On the second day of the Lunar New Year, Chen Zhou needed to go to his grandfather's house to pay New Year's greetings.

It's just that after receiving the red envelope, having lunch, and chatting with his grandfather for a while, Chen Zhou went home by himself.

After cleaning up the somewhat messy desks briefly, Chen Zhou thought about it, and there seemed to be no need to go out again in the past two days.

Then, this is the most suitable time.

Chen Zhou drank all the remaining half jars of spiritual elixir.

He then began to search for more knowledge about the Lagrange median value theorem, ready to find out the ins and outs of this theorem.

Let's start with the proof method.

"The Lagrangian median theorem can be proved by way of an auxiliary function:

It is known that f(x) is continuous in the closed interval [a,b] and is derivable in the open interval (a,b);

Then, construct the helper function g(x)=f(x)-f(a)-[f(b)-f(a)](x-a)/(b-a);

can be obtained, g(a)=g(b);

and because g(x) is continuous on [a,b], it is derivable in the open interval (a,b);

Therefore, according to Rohr's median theorem, there must be a little ε∈(a,b) such that g'(ε)=0;

From this, g'(ε)=f'(ε))-[f(b)-f(a)]/(b-a)=0;

deformed to obtain f(b)-f(a)=f'(ε)(b-a);

The theorem is proved. ”

This process is very simple, Chen Zhou understands it, but why to construct such an auxiliary function, and what is Rohr's median value theorem, he is confused.

Chen Zhou thought about it and immediately searched for the relevant concepts of Rohr's theorem.

"Rolle's median theorem is an important theorem in differential calculus, one of the three major differential median theorems, the other two are: Lagrangian median theorem, Cauchy median theorem..."

"It turns out that this guy also belongs to differential calculus..."

Chen Zhou continued to look at the description of Rohr's median value theorem, as well as the proof process.

This, the more he looked at it, the bigger his head became, Chen Zhou found that he didn't understand anything, he didn't know anything, and seeing a new theorem or lemma was a brand new knowledge.

Sure enough, 12 years of basic education is the real foundation...

Chen Zhou had a desire, and he strongly wanted to understand these theorem knowledge.

His curiosity was opened, and he no longer blindly studied for the college entrance examination.

At this time, Chen Zhou felt that this hidden mission seemed to become interesting.

He did not only focus on the Lagrangian median theorem and Cauchy's median theorem, which were mentioned in the task.

He began with the branch of the differential median theorem, which piqued his great interest, and started with the Rohr median theorem.

I have learned all about the proof process, the geometric significance, and several special cases.

For the Fermat lemma and the limit existence theorem mentioned in it, if he doesn't understand them, he will put them aside first, and just look at this Rohr median value theorem.

An afternoon was definitely not enough, and after Chen Zhou hastily settled dinner, he began to indulge again.

In order not to break this thirst for knowledge, Chen Zhou took out a new can of spiritual medicine and drank it all.

Like this, he only dared to do it before the start of school, the most suitable time.

This is not a joke, cultivating immortals requires the right posture, the right time, and the right place.

I have to say that with the assistance of the powerful head of the spirit elixir, he actually brushed all over the Rohr median theorem, to the Lagrange median theorem that he was already familiar with, to the only two Cauchy median theorems mentioned in the task, and then to the Taylor formula, Dab's theorem, and Lopida's law that he had never heard of.

Some of them understand and learn, and some of them are half-understood, and no matter how bad they are, they are familiar.

Chen Zhou finally understood why the hidden mission had to single out the Lagrange median theorem and Cauchy's median theorem.

Not only because they have a wide range of applications in the college entrance examination, but more importantly, the Lagrangian median value theorem is the core of the differential median value theorem, and other median theorems are the special cases and generalizations of the Lagrangian median value theorem, which is a bridge for the application of differential calculus and has extremely high research value in theory and practice.

The Lagrangian median theorem is a special case of Cauchy's median theorem.

It wasn't until dawn in the morning that Chen Zhou was called out for breakfast by Chen Jianguo, and he briefly broke away from the ocean of knowledge.

Chen Jianguo looked at his two deep eye bags, and was a little puzzled: "Xiaozhou, didn't you sleep well last night?"

Chen Zhou replied consciously: "Well, I didn't sleep... Well..."

After eating, Chen Zhou hurried back to the house to continue.

Although the strength of the potion had not yet passed, Chen Zhou was afraid that he would be sleepy, so he dried another can.

On the third day of the Lunar New Year, for a whole day, Chen Zhou did not take half a step out of the room except for going out of the house to eat.

For some physiological needs, he solved them in passing during the time of eating.

After another sleepless night, Chen Zhou fell deeply into this thirst for knowledge.

When he went to the toilet to wash his hands before eating, he found that he didn't seem to feel anything in the mirror, except that the dark circles under his eyes had worsened and the bags under his eyes had become bigger.

As for the spirit, it is extremely abundant!

Then go ahead...

On the fourth day of the Lunar New Year, after breakfast, another jar of spiritual medicine went down.

Chen Zhou's mind is now full of these differential median theorems, theorems to find limits, theta of finite incremental formulas, inequalities, functions, derivatives, and so on.

On the fifth day of the Lunar New Year, Chen Zhou looked at the last pot of potion left, and he hesitated a little: "Will it die suddenly? Is it really worth it for a hidden mission?"

After thinking about it, Chen Zhou calculated the time left before the superposition of the potions, and it seemed that there was not much left.

If he couldn't overcome this hidden task during the winter vacation, Chen Zhou felt that it was unlikely that he would be able to complete it with only three months.

As the new semester begins, it will definitely be bombarded, and there will be less and less time allocated to hidden tasks.

Glancing at the formula on the scratch paper again, he guessed that Lagrange, Cauchy, and Rolle must have spent a lot of effort to come up with these things, and maybe they would also cultivate immortals.

Thinking of this, Chen Zhou no longer hesitated, he raised his head and dried this last can.

Whether the cultivation of immortals can be achieved this time depends on the last wave.

The pen in his hand almost kept writing down his thoughts on scratch paper, and then corresponded to these theorems to verify his ideas.

Thoroughly understand every proof process, and know every application example by heart.

Let's go back and sort out the internal connections of these theorems.

"The Lagrangian median theorem is a generalization of Rohr's median theorem, and at the same time a special case of Cauchy's median theorem, a weak form of Taylor's formula (first-order expansion)..."

The days are short and the nights are long in winter.

But for Chen Zhou, there is no day or night, he only feels that one day passes too quickly.

He even felt that he had just eaten breakfast, so why did he eat breakfast again?

After another night of fierce fighting, at 7 o'clock in the morning on the sixth day of the Lunar New Year, Chen Zhou finished breakfast and continued to return to the house and sit down.

He sorted out the scratch paper he had written these days.

Chen Zhou has almost learned these differential median theorems.

He even learned a lot about advanced mathematics.

But he wondered why the system hadn't decided that he had completed the hidden mission.

While cleaning up, Chen Zhou looked at the proof process of the Lagrangian median theorem written on the first day, and couldn't help but smile slightly.

The logical order of this, he has now figured it out.

Because it is necessary to apply Rolle's median theorem to prove the Lagrange median theorem, a constructor is needed to satisfy the conditions of Rolle's median theorem, and the constructed function is not the only one, as long as the conditions of Rolle's median theorem can be satisfied.

Thinking of this, Chen Zhou picked up the pen and began to try to construct a new function to prove the Lagrangian median value theorem.

"Let F(x)=f(x)-[f(b)-f(a)]x/(b-a), because the function..."

“... So F(x) in..."

“... F(a)=f(a)-[f(b)-f(a)]a/(b-a)..."

“... Then F(a) = F(b), so that F(x) satisfies the three conditions of Roll's theorem.

Therefore, it is proven. ”

The moment Chen Zhou finished writing, the voice of the system sounded in his head.

"Congratulations to the host!

Chen Zhou didn't hear the rest of the words, the strength of the spirit medicine was over, he tilted his body, fell on the bed, and fell asleep.