On the first day of the new year, Chen Zhou spent his time solving questions efficiently.

There are still 4 and a half cans of psychoactive potion left.

On the second day of the Lunar New Year, Chen Zhou needed to go to his grandma and grandpa’s house to pay New Year greetings.

However, after receiving the red envelopes, having lunch, and chatting with his grandparents for a while, Chen Zhou went home by himself.

After briefly tidying up the messy desks, Chen Zhou thought for a while and realized that there seemed to be no need to go out in the past two days.

Then, this is the most suitable time.

Chen Zhou drank the remaining half can of psychoactive potion.

Then, he began to search for more knowledge about Lagrange's mean value theorem, preparing to figure out the ins and outs of this theorem.

Let’s start with the proof method.

“Lagrange’s mean value theorem can be proved using auxiliary functions:

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

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

It can be obtained that g(a)=g(b);

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

Therefore, according to Rolle's mean value theorem, there must be a point ε∈(a,b) such that g'(ε)=0;

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

Deformed to f(b)-f(a)=f'(ε)(b-a);

The theorem is proved."

This process is very simple, and Chen Zhou understands it, but why it is necessary to construct such an auxiliary function and what Rolle's mean value theorem is, he is at a loss.

Chen Zhou thought for a while and immediately searched for related concepts of Rolle's theorem.

"Rohr's mean value theorem is an important theorem in differential calculus and one of the three major differential mean value theorems. The other two are: Lagrange's mean value theorem and Cauchy's mean value theorem..."

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

Chen Zhou continued to look at the description of Rolle's mean value theorem and the proof process.

The more he looked at this, the more he became confused. Chen Zhou discovered that he didn't understand anything and knew nothing. Seeing a new theorem or lemma was a brand new knowledge.

Sure enough, the twelve-year basic education is really basic...

Chen Zhou felt a desire to understand these theorems.

His thirst for knowledge was opened up, and he no longer just studied for the college entrance examination.

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

He not only focused on Lagrange's mean value theorem and Cauchy's mean value theorem mentioned in the task.

He started with the differential mean value theorem, a branch that aroused great interest in him, and started with Rolle's mean value theorem.

Understand the proof process, geometric meaning, and several special situations.

Regarding the Fermat's Lemma and the Limit Existence Theorem mentioned in it, if he didn't understand them, he put them aside first and simply looked at Rolle's Mean Value Theorem.

One afternoon was definitely not enough. After Chen Zhou hastily finished dinner, he began to indulge again.

In order not to break this desire for knowledge, Chen Zhou took out a new can of psychoactive potion and drank it in one gulp.

He would only dare to do this at the most suitable time before school starts.

This is no joke, cultivating immortality requires the right posture, the right time, and the right place.

It has to be said that with the help of such a powerful psychotropic agent, he went from Rolle's mean value theorem to the familiar Lagrange mean value theorem in one night, and then to the only Cauchy mean value theorem mentioned in the task.

, and then there were Taylor's formula, Darboux's theorem, and L'Hourbid's law, which he had never heard of before. He actually went through them all.

Some have understood and learned, and some have only half-knowledge, or even worse, just look familiar.

Chen Zhou also finally understood why the hidden task had to single out Lagrange's mean value theorem and Cauchy's mean value theorem.

Not only because of their wide application in the college entrance examination, but more importantly, Lagrange's mean value theorem is the core of the differential mean value theorem, and other mean value theorems are special cases and extensions of Lagrange's mean value theorem.

, it is a bridge for the application of differential calculus and has extremely high research value in theory and practice.

Lagrange's mean value theorem is also a special case of Cauchy's mean value theorem.

It wasn't until dawn in the morning when Chen Zhou was called out to have breakfast by Chen Jianguo that he briefly escaped from the ocean of knowledge.

Chen Jianguo looked at the deep bags under his eyes and was a little confused: "Xiaozhou, didn't you sleep well last night?"

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

After eating, Chen Zhou quickly went back to the house to continue.

Although the potion was still powerful, Chen Zhou was afraid that he would become sleepy, so he drank another pot.

On the third day of the Lunar New Year, Chen Zhou never took a step out of his room except for eating.

For some physiological needs, he would take care of them during the meal period.

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

When he went to the toilet to wash his hands before eating, he noticed that he didn't feel anything in the mirror except that the dark circles under his eyes were aggravated and the bags under his eyes were a little bigger.

As for the spirit, it is extremely full!

Then go ahead...

On the fourth day of the Lunar New Year, after breakfast, I drank another can of psychoactive potion.

Chen Zhou's mind is now filled with these difficult-to-understand things such as the differential mean value theorem, theorems for finding limits, theta of finite increment formulas, inequalities, functions, derivatives, etc.

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

After thinking about it, Chen Zhou calculated the time left for stacking the previous potions, and it seemed that there wasn't much time left.

If he couldn't conquer this hidden task during the winter vacation, Chen Zhou felt that it would be impossible to complete it in the only three months he had.

As the new semester begins, there will definitely be a lot of bombardment, and the time that can be allocated to hidden tasks will become less and less.

Taking another look at the formulas on the draft paper, he guessed that Lagrange, Cauchy, and Rolle must have spent a lot of effort to come up with these things, and they might have become immortals.

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

Whether the cultivation of immortality can be successful this time depends on the last wave.

The pen in my hand is almost constantly recording my thoughts on the scratch paper, and then I will correspond with these theorems to verify my ideas.

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

Let’s go back and sort out the inner connections of these theorems.

"Lagrange's mean value theorem is a generalization of Rolle's mean value theorem, and it is also a special case of Cauchy's mean value theorem. It is a weak form (first-order expansion) of Taylor's formula..."

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

But for Chen Zhou, there was no day or night. He just felt that one day passed too quickly.

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

After another fierce battle all night, at 7 o'clock in the morning on the sixth day of the Lunar New Year, Chen Zhou finished his breakfast and went back to the house to sit down.

He sorted out the scratch papers he had written over the past few days.

Chen Zhou has almost learned these differential mean value theorems.

He even knows a lot about advanced mathematics.

But he was wondering why the system hadn't determined that he had completed the hidden mission.

While cleaning up, Chen Zhou looked at the proof process of Lagrange's mean value theorem that he wrote on the first day and couldn't help but smile.

He has now fully understood the logical sequence here.

This is because Rolle's mean theorem needs to be applied when proving Lagrange's mean theorem, so a constructor is needed to satisfy the conditions of Rolle's mean theorem. The constructed function is not unique, as long as it satisfies the conditions of Rolle's theorem.

can.

Thinking of this, Chen Zhou picked up his pen and began to try to construct a new function to prove Lagrange's mean value theorem.

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

"...so f(x) is..."

"...and f(a)=f(a)-[f(b)-f(a)]a/(b-a)..."

"...Then f(a)=f(b), so f(x) satisfies the three conditions of Rolle's theorem..."

"Therefore, it is proved."

The moment Chen Zhou finished writing, the system's voice sounded in his mind.

"Congratulations to the host! Completed..."
Chapter completed!