June 19, 20XX
Today was an extraordinary day in the Mushroom Kingdom. After the exhilarating Battle of Mushroom Raceways, where Mario and Sonic once again joined forces to thwart King Bowser and Dr. Eggman's plans, I found myself in the midst of a fascinating conversation with some iconic heroes.
As I walked through the aftermath of the race, I stumbled upon a gathering of new faces. Megaman, with his robotic stature as he introduced himself, caught my attention first. He mentioned something called Nintegaball, a power I was unfamiliar with but definitely intrigued by. The mention of Megaman drew the attention of Mario and Sonic, who were deep in conversation about their concerns.
Both Mario and Sonic expressed worries about the possibility of more antagonists emerging, such as the allies of King Bowser and Dr. Eggman who might seek to cause havoc. They specifically mentioned Team Mechas (Metallix's and Mecha Mario's other destructive henchmen who are just known as Mecha Sonic, Mecha Shadow, and Mecha Knuckles). Their presence was a cause for concern, as their robotic counterparts were known for their formidable strength.
In addition to this, Mario mentioned something called the Boos, who were also on the prowl, relentlessly seeking every power star artifacts in the Mushroom Kingdom. Their mischievous and ethereal nature always kept everyone on their toes.
However, our worries were somewhat alleviated as reinforcements arrived. Link, the valiant hero of the Hyrule Kingdom, and Samus Aran from Earth colony K-2L, joined the fray, ready to lend their strength to our cause.
A troubling revelation came to light during their conversations. Sonic mentions about the Chaos Emeralds, objects of immense power, were shrouded in a strange aura of pure negative energy. The implications of this discovery sent visible chills down to my new-founded friends who are Mobians, raising concerns about the potential consequences of such an ominous phenomenon.
As we delved further into our discussions, we learned that both King Bowser and King Wart had employed an increased number of Bullet Bills as part of their armies. These relentless projectiles always presented a challenge, but the thought of them becoming even more prominent caused us to consider new strategies.
And then there were the Shroobs, a race of alien beings with malicious intent, who had crossed paths with Mario and Luigi in the past. The mere mention of their name brought forth memories of battles fought and victories achieved.
Sonic, in particular, expressed deep concern about a potential threat known as Project Robotasizeing. The implications were dire, as it hinted at the possibility of innocent beings being transformed into robotic pawns. We vowed to investigate further and prevent such a catastrophe from occurring.
Meanwhile, it is an interesting note that more Mushroom Kingdom's military are scouting out to see if King Bowser's military forces (known as Shy Guys with Bullet Bill arsenals) are still in the areas nearby.
In the midst of our discussions, other important topics arose. Mario mentioned about the Crystal Stars, mysterious artifacts with untold powers.
Sonic added in the enigmatic capabilities of something called the Master Emerald, which seemed to be connected to a peculiar time vortex.
The mention of Fire and Ice Star Shrines piqued our curiosity, leaving us eager to explore the secrets they held.
Megaman, Sonic, and Mario altogether decide to find more clues on what their enemies are planning.
Just as we were engrossed in our conversation, unexpected arrivals joined our group. Goombella, a bright and knowledgeable companion, appeared to lend her support.
Soon after, a Mobian rabbit accompanied by six Mobian cats and another Mobian hedgehog, arrived on the scene.
Later then a female teenager (later found out that she has a friendship with Knuckles the Echidna).
Their presence was followed by another Mobian hedgehog.
Amidst the growing crowd, a male teenager with glasses stepped forward, adding his insights to our discussion. Each new arrival brought with them a unique perspective and the potential for new alliances and strategies.
Curiously, as I looked down at the ground, I noticed a newspaper lying there, seemingly forgotten. The headline mentioned a cloaked person stealing a museum's artifact.
With an underlying article discussing a museum exhibit centered around Earth's ancient Norman's Old English medieval history. While it seemed unrelated to our current predicament, I couldn't help but wonder if there was some hidden significance.
Math Notes: (filled with academic musings):
Cardboard: 50 cm by 20 cm
Maximize the derivatives
Volume= base x height
Rectangle with each corner (Area = x * x)
Rectangle's length: 50 cm
Rectangle's length without the corners: 50 - 2x
Rectangle's width: 20 cm
Rectangle's width without the corners: 20 - 2x
V= x (20-x)(50-x)
=x (1000 - 100x - 40x + 4x^2)
=4x^3 - 140x^2 - 1000x
v' = 12x^2 - 280x - 1000
Solve for x
4(3x^2 - 70x - 250)
-b -/+ square root b^2 - 4ac / 2a
70 -/+ square root (70)^2 - 4(250)(3) / 2(3)
=4.4... and 18.9...
4.4... is best answer
f(x)= 4x+7 / x-2
Quotient Rule
f '(x) = (x-2)(4) - (4x+7)(1) / (x-2)^2
0= 4x-8-4x-7 / (x-2)^2
= -15 / (x-2)^2
No intervals that is making it increasing, no critical points; first derivatives equal to zero.
Decreasing: (-infinity, infinity)
0= -15 / (x-2)^2
No critical points to be found
No increasing
-15 / (x-2)^2 = 0
f '(x)=0 Local Extrema
f "(x)=0 Inflection Points
f '(x) = -15 / (x-2)^2
Finding local extremas: f '(x)=0
Fractions can give critical points (extremas and inflection points)
Cannot divide by zero; no local extremas
Concave Up (slope is negative) and Concave Down (slope is positive)
No Local Extrema
No Concave Up or Down
Find the Inflection Points
No Inflection points
f '(x)= -15 / (x-2)^2
Chain Rule applied
f "(x)= -15 (-2) (x-2)^-3 (1) = 30(x-2)^-3
= 30 / (x-2)^3
30 / (x-2)^3 = 0
# / 0 = 0
So no inflection points
Find all asymptotes
You cannot take the dominator equal to zero
4x+7 / x-2
x does and cannot not equal to 2
So vertical asymptote is at x=2
Optimization Problems
f '(x) = 0
x, y
s=x+y
y is the constraint
60=x+y
60-x=y
P=xy
P=x(60-x)
P=60x-x^2
P '= 60 - 2x = 0
60=2x
So: x=30, y=30, s=60, p=900