Chapter 15: Who Says This Answer Is Wrong? [Seeking Monthly and Recommendation Votes]
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Wang Qingtao wrote on the blackboard quickly, finishing the solution to the problem in less than a minute.
“Given the function f(x) = eˣ + a x² – x, and f(x) ≥ 0.”
“When x ≥ 0, f(x) ≥ ½ x³ + 1. Find the range of values for a.”
“Everyone can try to solve this first.”
“Remember the twelve-character maxim I taught you: if your problem-solving approach isn’t clear, it will be very difficult to answer this question fully and accurately.”
“I’ll give you ten minutes.”
He turned around to face the students below, his face always wearing a smile, showing his expectation for the class.
After silently reading the problem’s conditions, everyone frowned but didn’t give up. They immediately took out their notebooks to start calculations, attempting to solve this major question from the city’s top high school.
The classroom, holding eighty or ninety students, suddenly became very quiet, with only the scratching sound of pens on rough paper.
Seeing this, Wang Qingtao didn’t idle either. After a pause, he stepped down toward the students.
When he passed Chen Lu and Li Wenhui, he deliberately stopped for a moment, lowering his eyes to check their solution steps.
From the slight shake of their heads, it was clear they hadn’t solved it.
As time ticked by, no one raised their hand; everyone remained focused, repeatedly trying and correcting.
One of the more extroverted students, who often joked with the homeroom teacher, spoke up.
“Teacher, can you give us a hint?”
“Zhou Junhao from the honors class didn’t ask for any hints,” Wang Qingtao looked toward the speaking student.
Mentioning this brought a bit of embarrassment to him. During lunch break, he had struggled a bit to solve these problems, hoping to boast to Cui Zhibang, only to be told that Zhou Junhao’s class had finished all of them in under an hour. Truly, he was the school’s most precious prodigy.
A genuine all-rounder never knew what it meant to be weak in any subject.
As this thought crossed his mind, his gaze naturally fell on Xu Ming, who was sitting against the wall on the left.
He recalled that Xu Ming had scored full marks on fill-in-the-blank and solution questions in the last practice test, with a problem-solving style similar to that of competition students.
His expectation rose considerably.
However, when he stepped to the left aisle and saw Xu Ming’s actions, his cheek twitched involuntarily.
His eyes landed on the desk, where Xu Ming was clearly reading a chemistry book.
Beside it lay a compulsory biology textbook.
Although he wanted to interrupt, he remembered instructions given by other subject teachers and reluctantly pretended not to see anything.
He let Xu Ming continue reviewing according to his own plan.
Then he returned to the podium and called on someone directly, hoping someone could solve it like Zhou Junhao.
“Chen Lu, do you have any solution ideas?”
“Not yet,” Chen Lu stood up and shook her head.
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Wang Qingtao waved his hand. “Sit down for now.” Then he turned to look at Li Wenhui.
“Class monitor?”
“Teacher, I’m still calculating,” Li Wenhui’s voice lacked confidence; he was uncertain too.
In such cases, Wang Qingtao didn’t hold much hope either. But just as he prepared to explain directly, he couldn’t help but call out Xu Ming’s name.
It was like buying a few lottery tickets—if you don’t scratch them all, there’s always a sliver of hope, even when you know the outcome.
“Xu Ming.”
…
Having experienced this a few times before, Qian Hong was already familiar with the routine.
As soon as the homeroom teacher’s eyes landed on Xu Ming, he guessed what was about to happen and nudged Xu Ming with his elbow to alert him.
“The teacher’s asking if you can solve the problem up there.”
“Oh.”
Xu Ming nodded and stood up, walking to the podium without a hint of hesitation.
Seeing this, Qian Hong was stunned for a moment.
He hadn’t expected Xu Ming to be so calm, completely out of his anticipation.
Quickly recovering, he gave Xu Ming’s back a thumbs-up—the signature gesture.
“Truly my deskmate.”
“Quite confident!”
From Xu Ming’s point of view, solving a math problem was nothing difficult; he thought it best to finish it quickly and return to review chemistry and biology.
So he simply went up.
Though he hadn’t yet carefully examined the problem content.
No doubt, Xu Ming’s move attracted the entire class’s attention.
After all, walking up without saying a word meant he was confident, having already solved the problem.
“Has Xu Ming already found the solution method?” Li Wenhui muttered inwardly, unwilling to accept this reality.
Chen Lu quietly watched Xu Ming, her eyes full of curiosity.
“Do you have a solution approach?” Wang Qingtao looked at Xu Ming on the podium, handing him the chalk as he asked.
“Let’s try,” Xu Ming replied calmly.
After this brief reply, he took the chalk and approached the blackboard, raising his eyes to examine the problem for the first time.
He didn’t delay long before, under everyone’s watchful gaze, he raised his hand and began listing formulas to solve it.
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“eˣ + a x² – x ≥ ½ x³ + 1, a ≥ – eˣ + (x³)/2 + x + 1/x² = h(x)”
…
“eˣ ≥ x²/2 + x + 1, at x = 0, left side = right side = 1”
…
“a ≥ 7 – e²/4”
With each mathematical symbol and formula written on the blackboard, the expressions on the students’ faces grew increasingly puzzled.
They simply couldn’t understand it.
Homeroom teacher Wang Qingtao first frowned, then seemed to realize something, his expression suddenly enlightened. Looking at Xu Ming, his eyes were like someone who had discovered a priceless treasure by the riverbank.
Just as he was admiring the solution steps, Xu Ming finished writing.
“Teacher, I’m done.”
“Wait a moment,” Wang Qingtao waved his hand at Xu Ming, then faced the class and announced the result.
“Xu Ming’s answer, a ≥ 7 – e²/4, is correct.”
The moment the words fell, the classroom erupted in a commotion, with many doubting the result.
Mostly because, even if they hadn’t found a way to solve it, it shouldn’t be so incomprehensible to the solution steps. This only meant the proof process seemed illogical.
Li Wenhui was the first to stand and express his opinion.
“Teacher Wang, Xu Ming’s proof process has many issues. This answer should be inaccurate, right?”
“I was just about to explain,” Wang Qingtao’s face showed delight; he was not surprised by Li Wenhui’s skepticism.
He gestured for Li Wenhui to sit, then stopped teasing the class.
“Xu Ming’s solution is indeed correct; however, he used the Taylor series, which is typically taught only in university.”
After speaking, he turned to Xu Ming and asked, “Have you studied the Taylor series?”
“I happened to come across it during the holidays and remembered it,” Xu Ming nodded in reply.
It wasn’t that he deliberately used a university-level formula to solve the problem. Since math had been elevated to a first-level discipline, the experience gained from standard exercises was limited. Only by involving university-level formulas and theorems could improvement be made.
Hence, he developed the habit of immediately applying university formulas when faced with problems.
At this moment, everyone in the class exchanged glances, utterly at a loss for how to describe their feelings.
…