Chapter 15: Who Said This Problem Was Solved Wrong?
Wang Qingtao’s board-writing speed was very fast; in less than a minute, he finished writing the solution to the problem.
“Given the function f(x) = e^x + ax - x^2, and f(x) ≥ 0.”
“When x ≥ 0, f(x) ≥ 1/2x + 1, find the range of a.”
“You can all try to do it first.”
“Remember the Twelve Maxims your teacher taught you. If your problem-solving思路 (approach) isn't clear, it will be difficult to solve this problem completely and accurately.”
“I’ll give you ten minutes.”
He turned around and looked at the students below, a constant smile on his face, expressing his anticipation for everyone in the class.
Everyone silently read the problem conditions. Although they frowned, they didn't choose to give up. They immediately took out their scratch papers to work on the calculations, attempting to solve this challenging problem from City First Middle School.
The classroom, which accommodated eighty or ninety people, was exceptionally quiet at this moment, with only the rustling sound of pen tips gliding across scratch paper.
Seeing this, Wang Qingtao wasn't idle either. After a pause, he walked down the aisles.
As he passed Chen Lu and Li Wenhui, he specifically paused for a moment, lowering his eyes to check their solution steps.
But judging from their subtle head shakes, they clearly hadn't figured it out.
As time ticked by, minute by minute, no one raised their hand. Everyone remained engrossed, constantly trying and failing.
A student with a more outgoing personality, who usually dared to joke with the Homeroom Teacher, took the initiative to speak up.
“Teacher, how about a hint?”
“Zhou Junhao from the Elite Class didn't need a hint,” Wang Qingtao looked at the student who spoke.
Mentioning this, he couldn't help but feel a bit ashamed. During lunch break, he had put some effort into solving these problems, intending to boast to Cui Zhibang. Instead, he was told that Zhou Junhao from their class had already finished solving all of them in less than an hour. One could only say he was truly worthy of being the school's most promising talent for Qinghua-Yan University.
A true Hexagon Warrior never knew what academic imbalance meant.
As his thoughts ran to this point, his gaze involuntarily fell on Xu Ming, who was sitting against the wall on the left.
He recalled how Xu Ming had gotten full marks on the fill-in-the-blanks and solution questions in the previous practice exam, and his problem-solving style was similar to that of a competition student.
His expectations immediately rose significantly.
However, when he stepped into the left aisle and saw Xu Ming's actions, a twitch ran through his cheek.
His gaze fell on the desk, where Xu Ming was actually reading a chemistry book.
Next to it was a required biology textbook.
Although he really wanted to interrupt the other party, he remembered what he had specifically told the other subject teachers. Reluctantly, he could only pretend not to see anything.
He allowed Xu Ming to continue reviewing his knowledge according to his own plan.
He then returned to the podium and directly called out names on the spot, hoping someone could solve it like Zhou Junhao.
“Chen Lu, do you have a solution approach?”
“Not yet,” Chen Lu stood up and shook her head, replying.
Wang Qingtao waved his hand, saying, “Sit down for now.” Then he turned to look at Li Wenhui.
“How about the class monitor?”
“Teacher, I'm still calculating,” Li Wenhui's voice was weak, and he, too, felt unsure.
Facing this situation, Wang Qingtao basically didn't hold much hope. However, just as he was about to explain it directly, he ultimately couldn't resist calling out Xu Ming's name.
It was like buying a few lottery tickets; if you don't scratch them all, you'll always have a lingering hope, even knowing the outcome.
“Xu Ming.”
...
Having gone through it several times before, Qian Hong already had experience in this regard.
One could say that as soon as the Homeroom Teacher looked over, he already guessed what was about to happen. So, he quickly nudged Xu Ming with his arm to inform him of the situation.
“The Homeroom Teacher is asking if you can solve the problem on the board.”
“Oh.”
Xu Ming nodded, then stood up and walked directly towards the podium, without a hint of hesitation.
Seeing this unfold, Qian Hong was also suddenly stunned.
He hadn't expected Xu Ming to go up so calmly; it was completely beyond his expectation.
He quickly snapped back to reality and, facing Xu Ming's retreating figure, once again gave his signature thumbs-up.
“As expected of my deskmate.”
“So confident!”
From Xu Ming's perspective, solving a math problem was incredibly simple for him. He just wanted to finish solving early and come back to review chemistry and biology.
So, he simply went up.
Even though he hadn't yet carefully read the problem's content.
And without a doubt, Xu Ming's actions at this moment attracted the attention of the entire class.
After all, going up directly without a word meant he was absolutely confident, having already solved the problem.
“Has Xu Ming already found the solution method?” Li Wenhui muttered to himself, unwilling to admit the truth.
Chen Lu quietly watched Xu Ming, her eyes full of curiosity.
“You have a solution approach?” Wang Qingtao sized up Xu Ming on the podium, asking as he handed him a piece of chalk.
Xu Ming calmly said, “I'll give it a try.”
After a simple three-word reply, he took the chalk and approached the blackboard. Only then did he raise his eyes to read the problem for the first time.
He didn't delay for long in this process. Soon, under everyone's gaze, he raised his hand to write out the formulas and solve.
“e^x + ax - x^2 ≥ 1/2x + 1, a ≥ -e^x + x/2 + x^2 + 1/x = h(x)”
...
“e^x ≥ x/2 + x^2 + 1, when x=0, left side = right side = 1”
...
“a ≥ 7 - e/4”
As mathematical symbols and formulas were written on the blackboard, the expressions of those below grew increasingly bewildered.
They couldn't understand it at all.
The Homeroom Teacher, Wang Qingtao, first frowned, then seemed to understand something, revealing a look of sudden realization. The way he looked at Xu Ming was like finding a priceless treasure by the river.
Just as he was admiring the solution steps, Xu Ming had already finished solving.
“Teacher, I'm done writing.”
“Wait a moment,” Wang Qingtao waved a hand at Xu Ming, then turned to the crowd and announced the result.
“Xu Ming's answer, a ≥ 7 - e/4, is correct.”
Almost the instant his words fell, the class immediately caused a stir, expressing doubt about this result.
Mainly because in everyone's opinion, even if they hadn't found the solution method for the time being, it shouldn't be that they couldn't even understand the solution steps, right? This only proved that the proof process was somewhat illogical.
Li Wenhui was the first to stand up and express his thoughts.
“Teacher Wang, Xu Ming's proof process has quite a few problems. This answer probably isn't accurate, right?”
“I was just about to elaborate,” a look of joy spread across Wang Qingtao's face; he didn't seem surprised at all by Li Wenhui's query.
After gesturing for Li Wenhui to sit down, he no longer kept them in suspense.
“Xu Ming's solution process is indeed correct. It's just that he used the Taylor Formula, which is only taught in university.”
After saying this, he turned to look at Xu Ming and asked, “Have you studied the Taylor Formula?”
“I happened to flip through it during the holidays and just remembered it,” Xu Ming nodded and gave this reply.
It wasn't that he deliberately used university-level formulas to solve the problem. The main reason was that after Math had been upgraded to Level 1, the Subject Experience Points he could gain from normal practice problems were limited. Only when involving university-level formulas and theorems would it improve.
This led him to develop the habit of using university-level formulas first when encountering a problem.
At this moment, everyone in the class looked at each other in dismay, completely at a loss for how to describe their current feelings.
...
(End of Chapter)