Yu Xiangnan looked worried.
Suddenly, two sheets of white paper and a black pen floated down from the sky.
Thank you, that's so thoughtful.
With things having come to this, Yu Xiangnan had no choice but to force himself to calm down and quickly recall the relevant knowledge.
Given the function f(x) = x^3 - 3ax^2 + 3bx, the equation of the tangent line to its graph at x = 2 is 12x + y - 32 = 0 ......
For the first question, we first differentiate the function f(x) and get f^\prime(x) = 3x^2 - 6ax + 3b.
Given the equation of the tangent line 12x + y - 32 = 0, it can be transformed into y = -12x + 32, with a slope of -12.
Since the slope of the tangent line to the function at x = 2 is equal to the derivative at that point, f^\prime(2) = -12, which is 3×2^2 - 6a×2 + 3b = -12 ①. Also, since the point (2, f(2)) is on the tangent line, substituting x = 2 into the equation of the tangent line gives y = 8, which is f(2) = 8, which is 2^3 - 3a×2^2 + 3b×2 = 8 ②.
Solve the system of equations ① and ② simultaneously to find a = 3 and b = 4.
Continuing with the second question, from the first question we know f(x) = x^3 - 9x^2 + 12x, f^\prime(x) = 3x^2 - 18x + 12. Let f^\prime(x) = 0, and find the extreme points of the function...
After finding the extreme points, analyze the monotonicity of the function to determine the maxima and minima. Calculate the maxima and minima to determine the range of values for k.
Congratulations, you answered correctly.
Yu Xiangnan took a deep breath; he felt completely overwhelmed.
She got it wrong twice in the middle, but luckily she got it right on the last try.
Thanks to the intelligent barrier, her rusty brain was able to regain some activity and recall the happy times of her high school life.
"Now we enter the second trial..."
The surrounding scenery changed once again.
As soon as Yu Xiangnan opened her eyes, she felt a wave of heat all over her body, and the blinding light made her squint.
After regaining her senses, she realized that she was in a blacksmith's shop.
The walls were covered with all sorts of tools: sledgehammers, pliers of various shapes, and bellows blackened by smoke. Scraps of iron and glowing red coals were scattered on the ground, and a pungent smell of burning filled the air.
The intense heat emanated from the roaring furnace.
Yu Xiangnan quickly noticed the game rules written on the wall:
Welcome to the Mighty Blacksmith Shop!
The following is a detailed explanation of the rules.
Resource Acquisition: At the start, players possess a certain amount of basic metal materials and a limited amount of coal fuel. Inside the blacksmith's shop is a mysterious ore mining spot. Players can mine there periodically, with a chance to obtain rare ores, but each dig carries the risk of failure and potential damage to tools.
The blacksmithing process involves placing metal materials into a high-temperature furnace and controlling the heat according to the material's properties (by adjusting the amount of coal added using a heat control bar). If the heat is too high, the material will melt and become unusable; if it's too low, the forging requirements cannot be met. Once the appropriate temperature is reached, the material is removed and hammered on an anvil with varying force and rhythm to shape it.
Different weapons have different requirements for striking patterns. For example, swords need to be struck evenly on the blade, while axes require more emphasis on the axe blade.
Quest System: Forge specific weapons according to order requirements; completion unlocks new forging recipes and advanced tools. Side quests include repairing damaged blacksmith facilities; completion rewards resources or special buffs.
Time Limit: The blacksmith shop has fixed business hours. Players must complete orders and tasks within the specified time, otherwise they will face reputation loss and a reduction in orders.
(Note: There are other blacksmiths in the town where the blacksmith's shop is located, and they are also competing for orders and resources. Players need to improve their forging efficiency and weapon quality to stand out from the competition and win more customers and rewards.)
Forging and blacksmithing?
Yu Xiangnan checked the resources at hand.
There are not many basic metal materials, and coal fuel is also limited.
Her gaze swept toward the mysterious ore excavation site in the corner, and she decided to try her luck first.
There are no forging quests yet, so let's prepare enough materials and fuel first, so we won't have to keep going down into the mines.
Yu Xiangnan picked up his tools, got ready, and went down into the mine.
During the initial excavation, the excavation site began to shake violently without warning, and the first excavation attempt failed, with the tools showing significant wear and tear.
Unconvinced, Yu Xiangnan quickly adjusted his strategy and went down into the mine again.
This time, they successfully unearthed a rare ore that emitted a faint glow, and also harvested a considerable amount of fuel.
bite--
New quest: Forge a standard longsword. (Countdown: 2 hours)
Yu Xiangnan quickly selected suitable metal materials and put them into the high-temperature furnace, keeping a close eye on the temperature.
When adjusting the amount of coal supplied, the materials can be ruined if not handled carefully.
After two failures, the temperature was close to the right range, and finally, at the perfect moment, the red-hot metal was removed and placed on the anvil.
As she was agonizing over the forge and tools, wondering where to begin, a semi-transparent humanoid figure slowly emerged.
Without uttering a word, the man walked straight to the blacksmith's workbench, skillfully picked up an iron block, placed it on the furnace to heat it, and his movements were fluid and graceful.
Clang—Clang—
Seeing the man's focused expression as he forged the iron, Yu Xiangnan instantly understood that this was probably a beginner's tutorial, teaching her how to forge a sword.
A moment later, the person turned into a wisp of light and disappeared.
With the tutorial in hand, Yu Xiangnan was full of enthusiasm. After all, this kind of frame-by-frame tutorial game doesn't require much brainpower, which is much better than math problems, which are a lifelong nemesis.
She rolled up her sleeves and, following the required striking pattern of the sword, struck again and again.
During the process, the metal exhibited minor flaws several times, but she adjusted them in time thanks to her control over the force and rhythm.
Side quests will also be released during the forging process.
The blacksmith's bellows was damaged, affecting the blowing efficiency, so Yu Xiangnan temporarily stopped his work and collected materials to repair the bellows.
Upon completion, additional coal fuel was also obtained.
As time passed, the blacksmith's shop was about to close. Yu Xiangnan quickened his pace and finally finished forging the longsword before the deadline.
[Testing in progress...]
[After inspection, the weapons were found to be of superior quality.]
The longsword was successfully forged in the second trial, and the trial automatically ended. Since a new trial lay ahead, no further progress was made.
The skeleton says that if players wish to continue participating in the trial, they can apply to challenge the second trial again after obtaining the Gold Badge. Players can choose to exit the trial at any time, and if they successfully complete it, they can unlock hidden achievements that further enhance the power of the Gold Badge.
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