usenovelgroup Chapter 23 Even if a demon is stupid, can he still not learn calculus?
Chapter 23 Even if a demon is stupid, can he still not learn calculus?
Given function:
f(x) = e^x - ln(x+1) - ax² - bx - cf(x) = e^x - ln(x+1) - ax² - bx - c, where a, b, c ∈ Ra, b, c ∈ R, and satisfying:
（Ⅰ）x=0x=0是 f(x)f(x)的极小值点，且 f(0)=0f(0)=0；
(II) The tangent line to the curve y=f(x) at x=0 is perpendicular to the line y=2x.
(1) Find the values of a, b, ca, b, and c;
（2）证明：当 x>0x>0时，f(x)>0f(x)>0恆成立；
（3）设 g(x)=f(x)−x33g(x)=f(x)−3x3，证明：存在唯一的 x0∈(0，1)x0∈(0，1)，使得 g(x0)=0g(x0)=0，并求 x0x0的近似值（精确到 0.01）；
(4) For any n∈N∗, prove the inequality:
...
After seeing the problem on the blackboard, Raphael and Damian immediately began to write furiously and start solving it.
Time passed slowly.
After five hours of hard work, Damian was the first to solve the problem.
After two more arduous sessions, Raphael finally solved the problem.
"How...how is this possible?!"
Looking at the smooth and elegant solution process on the blackboard to the right, Raphael was shocked, and for the first time, his handsome face showed an expression of utter astonishment.
Even when he personally witnessed Damian ascending the Ladder of Will, he was not as shocked as he was now.
After all, being able to do math problems or not are two completely different kinds of devils.
"Question 2, trigonometric functions..."
Then Sean wrote a second question on the blackboard.
Without giving it much thought, Raphael picked up the chalk again and immersed himself in solving the problem.
After several hours of hard work, the two of them successfully solved the second trigonometric function problem.
Following that were the third question on planar vectors, the fourth on geometric sequences, the fifth on spatial vectors, the sixth on spatial geometry, and the seventh on conic sections...
As time passed, the new demon students in the auditorium were completely stunned by Damian and Raphael's problem-solving journey.
Are they... really in the same year as us?
They were able to arrive at the answers to these profound math problems that they couldn't even understand in such a short time.
It's absolutely magical!
After a whole period of weeping.
"Huff, huff..."
Raphael stood in front of the blackboard with chalk in his hand, panting heavily. His forehead was covered in sweat, and he looked like he was about to collapse from exhaustion.
For the devil, physical strength may be plentiful, but mental strength is not so certain.
Doing math problems for an hour straight is just as exhausting for any demon as carrying a heavy load for a whole day of weight training and running.
In terms of intellect, I may not be inferior to Damian; in fact, he might be even more incompetent than me right now!
The moment the thought crossed his mind, Raphael secretly glanced toward Damian's location.
The black-haired boy stood straight in front of the blackboard. Although there was sweat on his face, his expression showed no signs of fatigue. Instead, it gave off a sense of excitement and anticipation.
Those dark eyes were even more piercing, sparkling with an astonishing brilliance.
Damian: "It's been a long time since I've done math problems. It's quite enjoyable to do them occasionally."
He...he's still excited!
Raphael was so shocked that he almost dropped the chalk in his hand.
Just then, Sean wrote a brand new problem on the blackboard.
已知函数 f(x)=ln(1+x)xf(x)=xln(1+x)，其中 x>−1x>−1且 x≠0x=0，补充定义 f(0)=1f(0)=1。
....（4）计算定积分∫01ln(1+x)x dx∫01xln(1+x)dx，结果保留为无穷级数形式。
This is a calculus problem that even graduates of the College of Babylon might not be able to solve!
Has Sean gone mad? He's actually giving calculus to first-year students!
Raphael was utterly incredulous; how could this problem possibly be solved...?
rustling.
Just then, the sound of chalk scraping against the blackboard came from the right.
No way...
The blond boy stiffly turned his head and saw Damian effortlessly solving the math problem, muttering to himself.
Although the black-haired boy didn't make a sound, Raphael, who had learned lip reading, immediately understood the meaning of the sentence.
"Even if a demon is stupid, can't he learn calculus?"
Raphael: "..."
Was he...was he mocking me?!
Even a devil can't be stupid enough to fail at calculus... Even a devil can't be stupid enough to fail at calculus...
These words echoed repeatedly in Raphael's mind, dealing him at least ten thousand blows to his spirit.
I absolutely have to solve this problem!
A flame seemed to ignite in the blond boy's deep blue eyes as he desperately tried to figure out how to solve the problem.
It seemed as if wisps of steam were rising from his head due to brain overload.
But as the old proverb from hell goes, "A demon in a madness can do anything except solve a math problem."
No matter how hard Raphael racked his brains, searching through all the knowledge of calculus he had memorized, he still couldn't come up with a solution to the calculus problem Sean had written on the blackboard.
So when Damian once again filled the entire blackboard with a solution so fluent it could be called beautiful, Raphael's blackboard remained empty.
The outcome is decided.
boom!
Seemingly due to excessive mental exertion, the blond boy collapsed to the ground, drawing gasps of surprise.
"Wow, as expected of Damian, he actually managed to exhaust Raphael to the point of fainting in the math competition."
"He can even solve calculus problems? Is he really a freshman?"
"Bloodline, willpower, knowledge, appearance—he lacks nothing!"
"By the way, did you guys hear what Damian said in front of the blackboard just now...?"
The demons whispered amongst themselves, their eyes filled with admiration as they looked at the black-haired boy on the platform who was anxiously examining the unconscious Raphael.
Exceeding someone's own level will attract jealousy.
But if you surpass it by too much, then what you will receive is utter awe and worship.
...
When Raphael awoke from his coma, he saw the wet towel on his forehead and the concerned black-haired boy beside him. He gritted his teeth, knowing that it was the other boy who had taken care of him while he was unconscious.
This time, however, he simply couldn't bring himself to say that Damian was trying to curry favor with him to win.
The three previous resounding defeats had completely shattered the arrogance Raphael had accumulated over the past eighteen years of smooth sailing, invincibility, and dominance over all demons.
"Thanks."
Raphael uttered the words through gritted teeth, then turned his gaze to the smiling magician beside him.
"Teacher Sean, may I suggest the topic of the fourth duel first?"
"Sure, but it depends on Damian's opinion," Sean said.
"Wait, I've already won three matches, plus the one Raphael conceded earlier. Logically, with seven wins and four losses, I should have already won and shouldn't need to duel anymore," Damian quickly said.
Although he found these competitions quite interesting, he worried that a pure physical contest would expose his weakness and make him look weak.
"That's true, but once the Seven Deadly Sins trial begins, it cannot be stopped, so the duel must continue," Sean explained seriously.
Just kidding, something this fun is something he would never stop doing without this rule.
"For the fourth duel, I hope it will be a puppet duel," Raphael said slowly.
A magic puppet?
Oh no, I don't even know what that is!
Damian's lips twitched slightly, and he suddenly had a bad feeling.