37: Chapter 37 Methodology

Jiang Yao followed Jiang Lin into his bedroom.

Jiang Lin's room was very small and piled with things, making it feel somewhat cramped just to turn around.

The desk lamp on the study table was still lit, emitting a warm yellowish glow.

The desktop was extremely tidy, with no extra books or test papers; in the center lay only a folded piece of A4 paper, on which were drawn some geometric topological figures that Jiang Yao could not understand.

And beside that piece of paper quietly lay a book titled "Advanced Mathematics (Physics Category) Volume 1".

Jiang Yao's gaze paused for a moment on that yellow book cover.

Advanced Mathematics.

She certainly knew this was not something within the high school mathematics system.

The competition class of Jiangcheng Middle School would dabble in some calculus, but non-competition students did not study this.

Yet, with this book placed on Jiang Lin's desk, she surprisingly didn't feel it was out of place at all.

After all, he had just taken the Jiangcheng first mock exam and crushed the competition to rank first in the entire city.

For someone who already stood at the pinnacle of the high school knowledge system, reading some college textbooks in advance during the winter break was simply the most reasonable thing in the world.

Jiang Lin pulled his wooden chair aside to make space.

"Sit."

Jiang Yao shook her head and said, "I'll just stand."

Jiang Lin didn't stand on ceremony or insist further; he pulled open the drawer, took out a black neutral pen and a stack of brand-new draft paper, and pushed them to the center of the desk.

Just at that moment, Zhang Xiufen walked in carrying two cups of freshly brewed hot tea, steaming hot.

"Take your time talking, I won't disturb your studying."

After saying that, she tiptoed out and considerately closed the bedroom door tightly.

The cups were placed on the desktop, and the water vapor curled upwards.

Jiang Lin reached out and pushed the cup closer to the edge inward by two inches to prevent it from being accidentally knocked over.

Jiang Yao finally unfastened the document bag she had been holding in her arms all along and pulled out that math test paper.

The corners of the test paper had become somewhat wrinkled due to repeated pinching and friction last night.

"It's this problem. I worked on it last night. I searched Xiaoyuan and Zuoyebang and found the steps, but there was one step where I didn't understand how its logical chain was connected..."

The test paper was completely unfolded and laid flat on the desktop.

Question twenty-one, a derivative parameter problem.

This problem was heavily circled by Jiang Yao with a red neutral pen, like marking a fortress on a map that had long resisted capture.

Beside it, two versions of the solution were densely written. One was in blue ink with neat handwriting, looking like it had been copied directly from a problem-searching app.

The other version was in pencil, which was interrupted halfway and heavily marked with a huge question mark.

Below the question mark were several traces left by an eraser.

Erased too hard, the surface of the paper was slightly fuzzy.

Jiang Lin didn't care about these; he just lowered his head to look at the problem.

Given the function:

f(x) = xlnx + x^2 + 1 - ax, x > 0.

(1) If for any x > 0, f(x) >= 0 holds, find the value range of the real number a.

(2) If the function f(x) has two different zeros x_1 and x_2, and 0 < x_1 < x_2, prove that: x_1x_2 < 1.

Jiang Yao used the tip of her pen to tap the line of blue handwriting in the middle of the second question.

"I solved the first question."

Seeming afraid that Jiang Lin would mistake her for a rookie who didn't even know the basic parameter separation method, she added an explanation.

"It's just separating a, letting g(x) = lnx + x + 1/x, and then finding the minimum value."

Jiang Lin looked at her slightly eager gaze, calmly nodded, and signaled her to continue.

After receiving confirmation, Jiang Yao took a deep breath and continued speaking.

"I also found the answer to the second question. Xiaoyuan Search has a version done using extremum point displacement, Zuoyebang has a version done using symmetrization construction, and someone in our school chat forwarded an analysis calculated hard using Taylor expansion. No matter which calculation process it is, I can generally follow its formula derivation step by step."

Speaking up to this point, her brows slightly furrowed, and her eyes revealed deep confusion and unwillingness.

"But I just don't know why, in this standard construction method analysis, when it's halfway through the proof, it suddenly asks me to compare the sizes of g(1/x) and g(x)?"

After finishing these words, Jiang Yao herself felt her throat somewhat dry and even felt a bit awkward.

Because the answer was clearly in her mobile phone photo album and could be pulled out to view at any time. The steps weren't particularly long either, just six or seven lines of formulas.

If she were willing, she could completely rote-memorize these six or seven lines of formulas, and whenever she encountered similar double-zero problems in the future, she could directly apply the templates of isomorphism or substitution.

After hearing her confusion, Jiang Lin reached out, took a brand-new blank piece of draft paper, and placed it between the two of them.

He didn't say another word and immediately start furiously writing a long string of exquisite analytical processes on the paper.

Instead, at the very top of the blank paper, he wrote a sentence that had nothing to do with mathematical formulas.

"What is this problem asking?"

Looking at this interrogative sentence on the paper that was practically like nonsense, Jiang Yao was slightly stunned.

This didn't look like the beginning of a science student's problem-solving steps. It rather looked like the soul-searching question uttered by an veteran criminal police officer when facing a tangled crime scene while reading a detective novel.

Or rather, like the beginning of some kind of interrogation.

It indeed looked like the beginning of some interrogation.

After finishing this sentence, Jiang Lin put down his pen, raised those eyes so profound they didn't look like an eighteen-year-old boy's, and looked at her peacefully and intently: "You know how to do the first question, so I won't explain it. No problem, right?"

Jiang Yao naturally had no problem; she hadn't come here to listen to nonsense anyway.

"Mm." She nodded hard, looking at him with eager eyes, waiting for him to toss out some advanced formula or ultimate move she had never heard of before.

"Then for now, let's ignore the construction in its answer," Jiang Lin said, his tone as flat as discussing today's weather. "First tell me, what does the given condition in the second question—the statement that the function f(x) has two different zeros x_1 and x_2, and f(x_1) = f(x_2) = 0—actually mean in mathematics?"

Jiang Yao lowered her head to look at the prompt on the test paper.

She felt this question was asked a bit too fundamentally, or even somewhat bafflingly. But she still answered honestly.

"It indicates that x_1 and x_2 are the abscissas of the two intersection points between the original function f(x) and the x-axis."

"Anything else?" Jiang Lin suddenly pressed further.

Anything else?

Jiang Yao, who never expected there would be such a follow-up question, couldn't catch up with her words for a moment.

However, her brain still spun at high speed, trying to scrape up any hidden clues related to zeros from the knowledge points, exam outlines, and question-type summaries that teachers had made them repeat over and over during this senior year of high school.

Is it the zero-point existence theorem?

Is it the preconditions for using l'Hôpital's rule?

Or is it the critical point of derivative sign change?

If she were usually in the classroom, those academic gods or the teacher on the podium seeing her get stuck would most likely give a hint at this time, or simply lead out the answer along her train of thought.

But Jiang Lin didn't.

He sat there, his hands naturally clasped and placed on the desk, his posture very relaxed, like an old Populus euphratica rooted in the Gobi Desert.

He didn't urge or rush to answer, nor did his eyes even wander; he just quietly waited for her to pierce through that layer of window paper herself.

This sudden calmness instead brought Jiang Yao an invisible sense of oppression, making her realize that she could no longer muddle through with those clichés and mechanical memories she usually used to perfunctorily deal with teachers.

Time passed minute by minute, and the alarm clock on the desk ticked away.

A full two minutes passed until Jiang Lin was certain that Jiang Yao truly hadn't established the underlying logical mapping at this node. Only then did he pick up his pen again and neatly write down the original function on the draft paper.

f(x) = xlnx + x^2 + 1 - ax.

Then below it, he wrote the equation: f(x) = 0.

"Since you've already used the parameter separation method in the first question," Jiang Lin tapped the equation with the tip of his pen, "then now, in this equality, re-express a."

Jiang Yao certainly had no problem with the algebraic transformation step. She immediately took the pen from Jiang Lin's hand and quickly wrote down beside it:

xlnx + x^2 + 1 - ax = 0.

Since x > 0, therefore: a = lnx + x + 1/x.

Writing up to here, she stopped her pen, raised her head, and looked at Jiang Lin with an uncertain gaze.

Jiang Lin didn't speak, just quietly looked at the formula she wrote out, encouraging her to keep thinking downwards.

Jiang Yao couldn't help feeling a bit guilty under his gaze. Looking at the formula she had written, a flash of the function she had defined in the first question suddenly crossed her mind, and she softly probed and said.

"Isn't this expression on the right side just g(x) in the first question? So when the condition says f(x) has two zeros x_1 and x_2, does it actually mean that the equation g(x) = a has two different real roots, and these two roots are just x_1 and x_2?"

Hearing these words, Jiang Lin finally nodded.

He took the pen and wrote on the paper: g(x) = lnx + x + 1/x.

Immediately following that, below this function, he drew a rectangular coordinate system and sketched a brief function curve within it.

With extremely smooth lines, it descended from the upper right side of the y-axis, then pressed out a valley bottom in the middle of the first quadrant, after which the curve smoothly rose towards the upper right.

This was a typical asymmetric U-shaped curve that looked like a checkmark.

After finishing the curve, he used the pen to draw a horizontal straight line y = a below the curve.

This horizontal line was like a knife, cutting across the waist of that U-shaped curve.

The tip of the pen gently tapped at the intersection point of the straight line and the left side of the curve.

"The abscissa corresponding to this point is the smaller root on the left, x_1."

Subsequently, the pen tip shifted horizontally to the intersection point on the right and tapped it.

"This is the larger root on the right, x_2."

Jiang Lin put down his pen, looked into Jiang Yao's eyes, and uttered a soul-striking sentence: "So, you see, the second question of this problem seemingly asks about the two zeros of the original function f(x), but actually, but actually, it is studying the abscissa properties of the two intersection points between this horizontal line y = a and the curve y = g(x)."

Jiang Yao was stunned for a moment, blankly looking at the extremely simple curve and that horizontal line on the draft paper.

When she looked at the analysis on her phone last night, the first line was clearly still f(x), with all kinds of derivation flying everywhere.

By the second line, it suddenly jumped to some complex constructed formula F(x) = g(x) - g(1/x), as if an insurmountable fault line had appeared in the middle.

But now, Jiang Lin had used merely an intersection diagram that a middle school student could understand to pin the two variables originally hidden in the fog clearly onto the coordinate axes.

She could intuitively see that x_1 was to the left of the valley bottom, while x_2 was to the right of the valley bottom.

Jiang Lin didn't give her too much time to reminisce; he used the tip of his pen to tap the conclusion that the second question of the problem finally required to be proven.

x_1x_2 < 1.

"Now, let's look at the objective. What is this statement equivalent to?" Jiang Lin asked.

Jiang Yao stared at this inequality for a few seconds.

After clarifying that both x_1 and x_2 were positive numbers, the transformation of the inequality became extremely simple.

"It's equivalent to..."

She wrote: x_2 < 1 / x_1.

Jiang Lin nodded.

"That's right, this is the objective of the problem: transforming the form of the product into a magnitude comparison of single variables. Now, look back at the image."

Jiang Lin pointed at the intersection point on the right with the tip of his pen: "Now, where is x_2?"

Jiang Yao looked over following the pen tip, her thinking gradually becoming clear: "On the right branch of the curve, the abscissa corresponding to the point on the right branch that satisfies g(x) = a."

"Then if you want to prove x_2 < 1 / x_1, and x_2 is on this monotonically increasing right branch, how should you use the monotonicity of the function to compare them?"

Jiang Lin tightened the core of the problem step by step.

Jiang Yao lowered her head to look.

On the right branch, g(x) was obviously monotonically increasing.

If it could be proven that: g(1 / x_1) > a, while g(x_2) = a.

That would mean on the right branch, the function value corresponding to the independent variable 1 / x_1 was higher than the function value corresponding to x_2.

Since the right branch slopes upward and is higher, it means the x-coordinate is also larger.

Then there must be: x_2 < 1 / x_1.

Jiang Yao slowly voiced her deductive thought process.

Her eyes grew brighter and brighter, just like someone groping in the dark who finally touched the light switch.

However, the more she spoke, the lower her voice suddenly became.

In the end, before she even finished the whole sentence, she stopped herself instead.

Jiang Lin did not immediately reply.

He turned his pen horizontally, letting the barrel gently press against the horizontal line representing y = a on the diagram, and then gently tapped x_2 on the right branch with the tip of his pen.

The room was quiet for ten-odd seconds.

Only then did Jiang Lin unhurriedly write down on the paper the formula that had caused Jiang Yao two hours of misery last night.

g(1 / x_1) - g(x_1).

"Since g(x_1) = a, to prove that g(1 / x_1) > a, you are essentially comparing the sizes of g(1 / x_1) and g(x_1). This is the origin of the so-called construction you saw."

Jiang Lin's voice was low and powerful, like a heavy hammer striking Jiang Yao's heart.

Jiang Yao stared blankly at this line of formulas.

Last night, when she saw the solution suddenly write let F(x) = g(x) - g(1 / x), she only felt that it was an inspired trick of some mathematical genius, a stroke of divine inspiration that a mortal like her could never think of in her lifetime.

But now, she knew that it did not fall from the sky at all.

It was forced out by the goal of the problem.

To prove x_2 < 1 / x_1, and since x_2 is in a monotonically increasing interval, in order to utilize monotonicity, 1 / x_1 and x_2 must be placed on the same monotonic interval for comparison.

To compare their x-coordinates, it naturally transforms into comparing their corresponding function values: the sizes of g(1 / x_1) and g(x_1).

At this moment, all the logical chains meshed together seamlessly.

The gears began to turn, emitting a deafening roar.

Jiang Yao subconsciously wanted to nod in agreement.

But halfway through her nod, she abruptly stopped.

She swallowed that breath back down.

Because this time, she wasn't in that state of half-understanding where she looked at the answer and felt what the other person said made some sense.

She truly, clearly saw how the foundation at the bottom of this building was laid. She saw the cause and effect.

Seeing that she seemed to have some realization, Jiang Lin immediately wrote down the analytical expression of g(x) on the paper.

g(x) = lnx + (x + 1) / x.

...

He did not write fast.

So that Jiang Yao's brain could keep up and digest it.

Jiang Yao initially wanted to nod and chime in, but later on, she simply stopped moving altogether. All her attention was tightly fixed on every line of characters under Jiang Lin's pen, staring at the origins and developments of those formulas.

Jiang Lin finished writing the last step, raised his head, and asked, "Alright, now that the logical chain has been transmitted to the final step, what has the problem turned into?"

Jiang Yao looked down along the handwriting that was neat to the point of being rigid, and replied, "Since the result of the construction is -2lnx_1, then to prove that g(1 / x) > g(x_1), we only need to prove that when x = x_1, -2lnx_1 > 0."

"Which means?"

"Which requires lnx_1 < 0, meaning x_1 < 1." Jiang Yao's train of thought flowed smoothly like ice-breaking spring water.

Jiang Lin nodded and threw out the final question: "Then, where does the conclusion of the final piece of the puzzle for the proof, x_1 < 1, come from?"

Jiang Yao immediately lowered her head to look at the result she had done for the first question.

She knew g(x) = lnx + (x + 1) / x.

In the first question, in order to find the minimum value, she had found its derivative.

g'(x) = 1 / (x + 1) - 1 / x^2.

Setting the derivative to zero to find the extreme points: 1 / (x + 1) - 1 / x^2 = 0.

Since x > 0, multiplying both sides of the equation by x^2 yields an extremely simple quadratic equation in one variable: x + x^2 - 1 = 0.

That is: x^2 + x - 1 = 0.

Solving this equation and discarding the negative root, the positive root obtained is: a = (√5 - 1) / 2.

This happens to be the golden ratio number.

Obviously, a < 1.

According to the relationship between the derivative and the extreme points, because g(x) is monotonically decreasing on the interval (0, a) and monotonically increasing on the interval (a, +infinity).

It is known from the problem that g(x) = a has two positive roots x_1 < x_2.

From the diagram drawn just now, it can be seen very intuitively that x_1 must be on the left side of the minimum point, which is the decreasing interval; x_2 must be on the right side of the minimum point, which is the increasing interval.

Therefore, there must be: 0 < x_1 < a < x_2.

Since x_1 < a and a < 1, according to transitivity, it naturally follows that: x_1 < 1.

Since x_1 < 1, therefore: -2lnx_1 > 0.

That is: g(1 / x_1) > g(x_1) = a.

Also, because it was discussed earlier, since 1 / x_1 > 1 > a, it shows that the point 1 / x_1 also falls on the increasing interval (a, +infinity). On this monotonically increasing right branch, since g(x_2) = a and g(1 / x_1) > a.

Therefore, the size relationship of the x-coordinates must be derived: x_2 < 1 / x_1.

Multiplying both sides by the positive number x_1, it is finally proved: x_1 x_2 < 1.

When Jiang Yao's brain followed this flawless logical chain all the way to the end in one breath, she suddenly felt that the menacing final-question problem before her eyes became very quiet.

It wasn't that she felt it had become simple. This problem was still very difficult.

Rather, she felt it had become quiet. It had been completely tamed.

Those formulas that were piled up messily in her mind last night, the fragmented steps copied from problem-searching apps, at this moment were like being placed back piece by piece into the positions where they were originally supposed to exist by a pair of extremely steady, large hands.

What does the problem want to prove?

What underlying mathematical relationship is this surface goal equivalent to?

To compare this relationship, who and who do we need to find as intermediaries?

Where on earth did that seemingly otherworldly construction come from?

Every step finally had a clear origin.

Jiang Lin gently placed his pen on the edge of the scratch paper.

"Those solutions you looked at last night probably omitted the process of analyzing the goal and translating the conditions beforehand, jumping straight from the middle, and showing you the construction formula right at the first step."

Jiang Yao nodded heavily, her eye sockets even faintly warming up: "Yes, that's right, they only wrote the proof process, not how the train of thought came about."

"Therefore, you didn't fail to understand the subsequent calculations; you just didn't know why it had to head in that direction," Jiang Lin said.

Honest advice is unpleasant to the ear.

Jiang Yao felt a bit uncomfortable hearing this.

As a top student at a municipal key high school, her pride made her instinctively want to retort.

But she could not retort, because Jiang Lin had hit the nail right on the head.

Last night she had indeed understood the subsequent derivative simplification calculations, and even today she could close her eyes and copy out an identical complete answer entirely from memory.

But if the answer was closed, if she was thrown into the Gaokao examination hall, and the problem was slightly modified to let her redo it herself, she still wouldn't think of comparing g(1 / x_1) and g(x_1).

She used to think that this meant she was too stupid and failed to think of the trick.

But now Jiang Lin used a piece of paper and a pen to tell her that this wasn't some trick suddenly appearing.

Rather, it was because right at the very beginning of solving the problem, she hadn't calmed down to firmly fix her eyes on the true goal of the problem.

Jiang Yao lowered her head, looking at the upright line of characters written by Jiang Lin at the very top of the scratch paper.

What is this problem asking?

She suddenly discovered with horror that in her three years of high school, her problem-solving practice was body-high, and the test papers she had done stacked up higher than many people's heights.

Yet she seemed to have rarely truly stopped before putting pen to paper to seriously ask herself this question.

Her usual path dependency was to look at the keywords in the problem stem.

Oh, it's always true, then separate parameters to find extreme values.

Oh, it's a derivative parameter problem, then classify and discuss to calculate roots.

Oh, it's a double-zero inequality proof, then try extreme point offset or isomorphism.

Then she would frantically search her brain's library for corresponding routine templates to apply.

This set of mechanized methods was certainly useful.

It was efficient enough to keep her ranking in the top thirty of the grade in the brutally competitive environment of Jiangcheng Middle School.

It was enough for all teachers to evaluate her as a seed player for 985 universities.

However, those four words, Tsinghua University and Peking University, seemed to be stuck in this invisible and intangible place.

This was no longer a question of whether she worked hard or not.

Rather, it was that sometimes she was too anxious to run forward, so anxious that she buried her head in the sea of problems, completely failing to see clearly which path she was actually standing on.

Jiang Yao's fingers holding the pen turned slightly pale, and she raised her head, asking in a very soft voice: "So Jiang Lin, this kind of exquisite construction has never been memorized by rote learning, right?"

Jiang Lin thought for a moment and gave a very objective answer: "Some can indeed be memorized. Many so-called problem-type ultimate moves are essentially routines summarized by predecessors. For test-taking speed, memorizing routines is not wrong."

Jiang Yao looked at him in confusion, as this seemed to contradict the method he had just taught.

Jiang Lin explained, "But constructions that are memorized are easy to forget and easy to use incorrectly."

He picked up his pen and tapped again with the tip on the rows of formulas on the scratch paper deduced backward from the goal.

"Only like this, a construction deduced step-by-step backward from the final goal, has its logical root system anchored in the conditions of the problem itself. This kind of thing that grows in your own brain will never be easily lost as long as you don't forget the basic axioms of mathematics."

Jiang Yao looked at that line x_1 x_2 < 1.

Then she looked at its equivalent x_2 < 1 / x_1.

Finally, her gaze lingered on g(1 / x_1) - g(x_1), which it naturally extended from.

She suddenly understood.

It wasn't just this problem that she understood.

The solution to a derivative final problem was just a drop of water in the ocean of mathematics.

It was something that shocked her even more, something that made her even more silent; she seemed to have touched its edge a little bit.

That was something about the essence of learning and the reconstruction of cognitive systems.

She folded the scratch paper full of derivation processes and put it back into the document bag along with the test paper.

Her movements were much slower than when she arrived, as if she was afraid of disrupting the fragile cognitive system that had just been established.

"Jiang Lin."

"Hmm?"

"Every time you solve problems now, do you always think from the very bottom level like this?"

She looked at the boy before her who had once fallen behind in her eyes.

Jiang Lin frowned slightly, thought about it for a moment, and said, "As long as it's difficult."

Jiang Yao was somewhat surprised.

Jiang Lin continued, "I can only say as much as possible. At least when encountering truly good problems, don't rush to look at the answer; try to think more about why."

Jiang Yao opened her mouth, wanting to ask: *Then how could you train your thinking to such a terrifying depth in such a short period of time?*

But when these words reached her lips, in the end, she still didn't ask them out loud.

She was a smart girl and knew that she wouldn't get the real answer by asking.

Some things required the precipitation of years or extremely high talent.

Just like the construction of this problem, if one only looked at Jiang Lin's current grades, it would always look like a miracle falling from the sky.

All she could do was go back and slowly digest the method she had just learned.

"Thank you, I've really troubled you today."

"You're welcome."