text:
The minimum value of the polynomial $x(x+1)(x+2)(x+3)$ is:
a) $0$
b) $9/16$
c) $-1$
d) $-3/2$
Please answer in detail.
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The minimum value of the polynomial $x(x+1)(x+2)(x+3)$ is:
a) $0$
b) $9/16$
c) $-1$
d) $-3/2$
Please answer in detail.
This question appears to be off-topic. The users who voted to close gave this specific reason:
Hint: If $$f(x)=uvw$$ then we get $$f'(x)=u'uw+uv'w+uvw'$$ $$f'(x)=\left( x+1 \right) \left( x+2 \right) \left( x+3 \right) +x \left( x+2 \right) \left( x+3 \right) +x \left( x+1 \right) \left( x+3 \right) +x \left( x+1 \right) \left( x+2 \right) $$
As you know, To find the minimum value of any function we differentiate it and then equate it to $0$. The reason for this is that:
The graph has horizontal tangent where the function has maximum or minimum value. So, in order to find the points where the function maximum or minimum values, first you need to find the derivative of the function and set the derivative is equal to zero and solve the equation for the points called critical points.
Now, to find the minimum value of the polynomial of your function, you can write it as:
$f(x) = x^4 + 6x^3 + 11x^2 +6x$
Differentiate with respect to $x$
$f'(x) = 4x^3 + 18x^2 + 22x +6$
Which can be written better as:
$f'(x) = (2x+3)(2x^2+6x+2)$
Now you have to equate $f'(x)$ in order to get minimum value.
$f'(x) = (2x+3)(2x^2+6x+2) = 0$
This means either $2x+3 = 0$ or $ 2x^2+6x+2 = 0$
Either $x = -3/2$ or $x=(-3 + √5)/2$ or $x=(-3 - √5)/2$
Now you can check values of $f(x)$ with these critical points and the one with the minimum value, will get you your answer.
The maxima/minima can be found by finding the derivative of the function and setting it to equal $0$. $$f(x) = x(x+1)(x+2)(x+3) \implies f(x) = x^4+6x^3+11x^2+6x$$ $$f'x = 4x^3+18x^2+22x+6$$ The derivative should be factored so it could be solved. $$\implies 2(2x^3+9x^2+11x+3)$$ $$\implies 2[(2x^3+3x^2)+(6x^2+9x)+(2x+3)]$$ $$\implies 2[x^2(2x+3)+3x(2x+3)+1(2x+3)]$$ $$\implies 2(2x+3)(x^2+3x+1)$$
Set $f'x = 0$
$$f'x = 0$$ $$2(2x+3)(x^2+3x+1) = 0$$ $$2x+3 = 0 \implies 2x+1 = -3 \implies \boxed{x = -1.5}$$ $$OR$$ $$x^2+3x+1 = 0 \implies \boxed{x = \frac{-3\pm\sqrt{5}}{2}}$$ The final step is carrying out the second derivative test and plugging in $x$. $$f''x = 12x^2+36x+22$$ The question asks for a minima, so $f''x > 0$, as the slope becomes positive past the inflection point.
$$f''-1.5 = -5$$ $$f''\frac{-3-\sqrt{5}}{2} = 10$$ $$f''\frac{-3+\sqrt{5}}{2} = 10$$
$\frac{-3\pm\sqrt{5}}{2}$ meets this criterion, substitute it in place of x in the original function. (Both of them give the same answer.)
$$y_{minimum} = \frac{-3\pm\sqrt{5}}{2}\biggr(\frac{-3\pm\sqrt{5}}{2}+1\biggr)\biggr(\frac{-3\pm\sqrt{5}}{2}+2\biggr)\biggr(\frac{-3\pm\sqrt{5}}{2}+3\biggr) = -1$$
So, the minimum value of the polynomial is $-1$, when $x = \frac{-3+\sqrt{5}}{2}$ or when $x = \frac{-3-\sqrt{5}}{2}$.


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subset: commoncrawl


text:
Judge strikes down part of Canada's terrorism law
OTTAWA, Oct 24 (Reuters) An Ontario judge struck down a section of Canada's Anti-Terrorism Act today, saying it violates freedom of religion and thought guarantees in the country's Charter of Rights.
Ontario Superior Court Judge Douglas Rutherford made the decision in the case of a Canadian who is charged with conspiring to carry out bomb attacks in Britain.
Rutherford said the clause in the law that limits the definition of terrorist activity to acts motivated by religion, politics or ideology is unconstitutional. He severed that clause from the rest of the Anti-Terrorism Act, which he left in place.
His ruling was in response to a constitutional challenge by Momin Khawaja, a Canadian Muslim who was the first person charged under the Anti-Terrorism Act, which became law in 2001 following the September 11 attacks on the United States.
Khawaja faces seven charges under the act that do not detail religious or ideological motivation but rather such concrete actions as conspiring with others to develop explosives.
Rutherford ruled that those charges should proceed, and rebuffed an argument by Khawaja's lawyer that the Anti-Terrorism Act is too broad and vague.
Khawaja, arrested in 2004, remains in jail and will proceed to trial on January 2. Seven British Muslims accused of being his co-conspirators are on trial separately in Britain.
One of them, Omar Khyam, told a court in London last month of his delight with the 2001 attacks on the World Trade Center in New York, and said Pakistanis and Afghans loved the attack's mastermind, Osama bin Laden.
One of the reasons the Canadian legislation had included the section on political, religious or ideological motivation was so that people with other motivation, for example labor union demonstrators, would not be labeled terrorists.
Rutherford said the effect, however, was to unfairly focus investigation on political, religious and ideological beliefs.
Reuters PDS VP0030


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Kenta Maeda
Takes loss against Giants
Maeda (7-4) took the loss against the Giants on Monday, giving up two earned runs on three hits over five innings, striking out four and walking two in a 3-2 defeat for the Dodgers.
Kody Hoese
Impresses in minor-league debut
Hoese went 3-for-3 with three doubles and an RBI for the Arizona League Dodgers on Monday.
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Pollock (elbow) took some swings off a tee Monday, Ken Gurnick of MLB.com reports.
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Seager (hamstring) played catch Monday, Ken Gurnick of MLB.com reports.
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Site Medical Director (San Rafael, CA)
Company: MDstaffers
Location: San Rafael
Posted on: June 13, 2019
Job Description:
To view more of our open positions, please visit To immediately speak with one of our staff, please call (866) 90-STAFF. MDstaffers is a proud member of National Association of Locum Tenens Organizations (NALTO) and National Association of Physician Recruiters (NAPR). Follow us on LinkedIn, Twitter, Instagram, and Facebook for new job postings.
Job Reference #13022
Keywords: MDstaffers, Santa Clara , Site Medical Director (San Rafael, CA), Executive , San Rafael, California
here to apply!
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Colby G.
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Michael H.
Michael is the most senior team member at over 15 months at Platte Valley! He's here to help you Sunday through Wednesday.
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Mick has been with us a few months but has always fit in like an elder. He's the newest addition to our management team. Come see him on Wednesday, Thursday, Saturday & Sunday.


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If you use the internet much, no doubt you were affected by yesterday's huge internet outage. If you didn't experience it like most if not all Telecel users then at least you somehow heard or read about it. Also, if your dealings were/are local, Telecel user or not, then you were still negatively impacted since most internet dependent services were unavailable as well, so were some of your friends anyway…
Why this massive internet outage? Techzim wrote about it here, but in summary the outage resulted from accidental breakages on both Liquid and TelOne's main fibre lines. Not to say that Liquid and TelOne share their main fibre lines, no. Liquid's main fibre line was cut by a tractor in South Africa while TelOne's was cut locally by a city council worker in Kuwadzana.
So both these main fibre lines just happened to be cut on the same day and right about the same time… sounds dodgy doesn't it???
Well, you not the only one who thinks that. Hon Supa Mandiwanzira, the 'new' minister of the new Information Communication Technology and Cyber Security ministry expressed the same sentiments, not in those exact words but yea something like that.
Not only is the timing of these two incidents suspect but the fact that even the redundancy lines were affected as well, not to mention the reasons for the breakages. So for starters, at Liquid they say it's a truck that cut the fibre connection, how deep or rather how shallow could a main fibre line be? Secondly, well, say that's not a problem since we could just switch to a redundancy line but no, no, the redundancy line was also cut by the same truck because guess what, both the active and the redundancy lines are running in the same direction!
How? Maybe someone who knows better can explain it to me in the comments section, otherwise to me for now it all sounds like storing your laptop's backup files in a folder within that very same laptop.
Anyway, Hon Supa wasn't in the least pleased by this and clearly he is not to convinced with these reasons being given by these Internet Access Providers (IAP). In his own words according to the Newsday he said:
This should not happen, there should be redundancy plans in place and I wonder why these did not kick in. If we find that these companies were not truthful with us, we will revoke their licences.
Now we wait for the investigations around the issue and see what happens. But then again, if these two companies' licences are revoked, then what? Seriously, then what? Remember, Liquid carries 81.4% market share of equipped international internet bandwidth while TelOne (second largest) takes up 15.1%… that's like 96.5% of all our bandwidth… gone.
See what monopoly or rather lack of competition does???
Quick NetOne, Telecel, Africom, Econet Airtime Recharge
Supa Collins Mandiwanzira is a Zimbabwean politician, entrepreneur, former government minister and former journalist. He is the former Minister of Information Communication Technology and Cyber Security in the Republic of Zimbabwe. Mandiwanzira is the founder of Zimbabwe's only privately owned radio station, ZiFM Stereo through... Read More About Supa Mandiwanzira
Kuwadzana is a high density residential location located in Zimbabwe's capital city Harare. Read More About Kuwadzana
Telecel Zimbabwe is a telecommunications services provider with most of its services in the mobile cellular network services. The company is owned jointly by the Empowerment Corporation (40%) and Vimpelcom (60%), one of the largest telecoms companies in the world. In Zimbabwe, Telecel is the... Read More About Telecel
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- Police Investigating Suspicious Incident On ElmwoodPosted 1 week ago
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Each modern house is done by wardrobes and this can be applied not just to keep outfits but different vital things similar to shoes or boots, bags etc. diy small closet organization sometimes become the biggest part household which in turn are not split up through homeowners.
Hold your room to generally dried since your outfits and different stuffs can not lasting longer if they're subjected by damp air. Keep the area is obviously in the temperature it should be. Since we realize that moist air influents form to grow up in your clothes.
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"This is an easy to use, easy to clean citrus juicer that I wish I had bought 25 years ago! It is versatile (will juice small fruit like limes or large fruit such as Meyer lemons and even relatively small grapefruit. There is an adjustment allowing a little or a large amount of pulp to go through with the juice. It is well made and requires only a small space for storage. I will be buying oranges by the bag to have fresh juice. Probably my favorite small appliance."
This machine does come with a 1000 mL pitcher, which means it could hold roughly three 12-ounce sodas. Fresh juice does not keep for long, so you typically wouldn't make lots of it in advance; however, if you are making juice for your family or for the next day or two, it is nice not to have to stop and empty the pulp. You can get through a lot of produce before you need to pause.
Auger-style juicers, sometimes referred to as masticating or cold-press juicers, crush and mash the produce. They're typically more expensive, and can also take some getting used to as the augers can jam when grinding tough fruits and veggies. For this reason, every one we tested has a reverse button. The upside with this style is they tend to leave more healthful and fiber-rich pulp in the juice.
A passionate writer and a blogger, a person who is a movie junkie and a crazy lover of pets of any size and kind. The Ocean combined with a warm breeze what really inspires and makes her smile. She loves writing just about everything that help people overcome an obstacle: marketing strategies, technological innovations, healthy living and human rights, freedom of opinions and much more…
Juicing is an amazing way to get loads of hard-to-come-by micronutrients and natural enzymes into your system, all while lessening the workload for your digestive system. Producing this juice means chopping, slicing, and squeezing your fruits and veggies into tiny little pieces to release to the most juice possible. When you are dealing with such small food particles, you will quickly find they have a strong tendency to get stuck in cracks and crevasses.
I love that this is more than just a juicer - you can make nut butter, pasta, and frozen sorbet, and so much more. So far, I have made a lot of frozen treats, all I do is put anything frozen in the tube, change out the juicing screen to the blank screen, an ta-da! Frozen sorbet, at the ready, the first time I tried it I made 3 different sorbets, frozen coconut milk, frozen strawberries, banana, and peach. Then I put in mixed berries too! It is so easy and fast, and tastes delicious, very similar to some of the water ice we get up the street - but so much healthier! Overall, I am very, very satisfied with this juicer!"
Another important consideration is where you plan to store your juicer and how easily you'll want to access it. While horizontal-masticating juicers can churn out high volumes of juice, you may not have space for them. Plus, they tend to weigh more so tend to be difficult to move from closet to counter. The small footprint of vertical juicers is ideal for smaller kitchens with limited counter space and can often tuck under cabinets. Lightweight centrifugal models are perfect if you need to move your juicer from storage space to counter. Just make sure whatever model you pick can handle the volume of juice you plan to make: smaller juicers equal smaller amounts of juice per batch.


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Shark Attack 3: Megalodon is the second sequel to Shark Attack, Starring John Barrowman and Ryan Cutrona, Directed by David Worth.
The film begins with a group of divers installing a power cable at the bottom of the Pacific Ocean, when a shark swims up and kills one of them. Six months later, lifeguard Ben Carpenter (from the first Shark Attack film) and his partner Esai drive out to sea to catch some lobster. While diving, Ben finds a broken power cable with a large shark tooth stuck in it. After he pulls out the tooth, he is caught by two other divers, and soon goes back to land. Later he posts a description of the tooth online, but as he cannot find a matching shark tooth on the internet, he names it "mystery shark". Natural history researcher named Cataline "Cat" Stone gets the message, finds him and looks at the tooth. She believes it to come from a huge prehistoric shark called Megalodon (Greek for "big tooth").
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Axis Scientific 22-Part Disarticulated Life-Size Human Skull
Providing you with another beautiful and useful model, here is the Axis Scientific 22 part disarticulated human skull. This model is great for hands on studying of the different bones of the skull individually. This model cannot be reassembled, as the detailed bones come supplied completely disarticulated.
The following bones are present:
- Parietal bone (left and right)
- Occipital bone
- Frontal bone
- Temporal bone (left and right)
- Sphenoid bone
- Ethmoid bone
- Vomer bone
- Zygomatic bone (left and right)
- Upper jaw (maxilla) with teeth (left and right)
- Palatine bone (left and right)
- Nasal concha (left and right)
- Lacrimal bone (left and right)
- Nasal bone (left and right)
- Lower jaw (mandible) with teeth
Axis Scientific provides models with accurate detail at a great price. This collection of anatomical models is definitely the answer for students and educators with a tight budget.
Made by Axis Scientific and backed by a 3 year warranty. Weighs approximately 2 lbs.
|Product FAQ
Q: Can this skull be articulated / assembled?
Q: What is the model made of?
Q: Who is this model made for?
Q: Will this include all the bones of the skull?
A: Yes, while you cannot assemble this model, if you were to line all of of the pieces up, it would create a complete skull.
Q: I have lost a piece of this skull. Do you have replacement parts?
A: Please contact support at 800-422-1134, we will be more than happy to see what we can do for you.
Q: Will I be able to view the bony landmarks on each piece of the bones?
A: Yes, this model is very detailed. You will be able to view all of the bony landmarks that occur on a human skull.
Q: Is this model numbered or labeled?
Q: Is this skull gender specific?
Q: Does this skull come with any teeth?
|Warranty Period
|3 Year Warranty
|Product Class
|Best Uses
|Medical Education, Patient Consultation
|Product Extras
|Product Manual


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|Phil Burpee is a carpenter and
farmer living north of Pincher Creek.
He keeps an eye on the world
from under the big Alberta sky.
Phil Burpee, Columnist
There was a time in the ancient of days, which is to say before three or four years ago, when conversation used to be a lively process of statement and conjecture and the give-and-take hashing-out of ideas and opinions. I might stop on Main Street to enjoy a bit of jaw-wag with a fellow citizen who is, say, a Wild Roser or some such. In the course of things, such a person might opine that the Earth is flat, or that Danielle Smith is sensitive to the plight of Alberta farmers, or some such other scrap of received wisdom. In olden times I would say something like - "Well now, I'm not so sure you've got that altogether right there, Cecil. Let's take a look here." And then I might go on to observe that my neighbour had just sent me a note from Australia, where he'd gone to visit his son, and he made no mention whatsoever of there being a big drop-off of any sort along the way. And when he'd looked out the window of the plane, the edge of the world looked decidedly curved, just like it might if you were flying around a big ball. I might go on to point out that Ms. Smith is so in hock to corporate backers that the chance of a farmer's lot improving under her whip hand were about as good as the proverbial snowball's chance in Fort Lauderdale. And away we'd go, tit-for-tat, this-and-that, hands a-waving - making conversation. Lovely. But not anymore.
Now there're more than two people in this conversation. Now there's a cluster of fleshy-palmed, Yoplait-slurping, thirty-something brats down in Silicon Valley muscling in on this most timeless of human activities. If I were a modern person, which apparently I am not, I would immediately curtail the sheer joy and plasticity and living wonder of this thing called conversation, and bring down the dread guillotine of disembodied judgment. No more to roll around on the mind's tongue the delicate assemblage of flavours and spiced nuances of thought and experience. No more the scintillation of cut and thrust. No more to relish the sight of my interlocutor pushing back his hat and scratching his head with a funny squished-up look on his face as a result of my colourful and convincing arguments. All this has been swept aside. There's a new bully on the block. I would now immediately disconnect my brain from the goings-on, whip out my smart phone, and utter those cold, stark words - "Let's google that."
Oh dear. At what other point in the evolution of human consciousness have three such unlovely little words so fundamentally shifted the very precepts of knowledge and understanding - of wisdom itself. For is not 'homo sapiens' so named as being ' man of wisdom'? And is not wisdom itself a thing hard-earned and hard-won? You can't buy it. You can't cheat it. You can't fool it. You can't pretend it. And you sure as goddamn well can't google it. It's a thing you can only grow into, and it is the hallmark of all maturity. A great tree is wise to the wind in that it has shed its weak and cross-wise branches, which would only anyway one day snap and cause sap to bleed. Grandmother Grizzly sees the quad approach down below, and wisely understands that she must gather her kin and move up the mountain. Elders of our species, until we began to diaper them and shuttle them off to La-La Land with their Jeopardy and their Jell-o, had historically developed the deep-lined faces and smoldering, implacable gazes of those for whom life has been slowly relinquishing its great mysteries - and with that, its greatest and most poignant and bittersweet beauties - evanescence, release, selflessness, loss, solitude, pain, touch, sharing, return, cycles, and the sublime ability to still love the foolishness of youth. And now we throw that all away, and relinquish our awareness to the Cloud - increasingly unable to articulate our own thoughts without the intercession of the Machine, and slowly succumbing to the bland collectivization of experience.
How do we learn things? It's an open debate. When I was in public school, apparently it was done through a process of variously being whacked upside the head with a yardstick, and being made to stand in the corner for long periods of time - oh, and maybe filling three or four blackboards with "I will not shoot spitballs in school", line after line after line. OK, that's one approach. Later on in high school, it seemed to be a matter of gorging on big long lists of facts and figures, and then being expected to regurgitate them on an exam in a room full of several dozen other hapless inductees. OK, that's another approach. But learning stuff is a tricky thing. Not to put too fine a point on it, but you can't learn to ride a horse until you've been piled off into the rocks a good few times. And you certainly can't learn how to speak as a toddler without making a whole bunch of funny noises along the way, before somebody finally figures out that you're trying to say that you "WANT THE RED BALL!" ("AH DO LAH BEEBEEE!"). I've long remembered a favourite line in a flick from the sixties or seventies (better google that) called 'The Prime of Miss Jean Brodie', with Maggie Smith. Her character works at a girls' school in England somewhere, and is having a heated discussion with a fellow faculty-member (her squeeze, as I recall). At one point he refers to her as a 'teacher'. She bristles, and retorts - "I am not a teacher. I am an educator, from the Latin e ducere - to lead out, as from the darkness." Beautiful. Knowledge and Understanding are to be drawn out, not jammed in.
Ignorance is truly the darkness. Knowledge is truly the light. Understanding is the slow-flowering mechanism by which we journey from one to the other. It requires our full attention. Conversation, wherein learning so often occurs, entails speaking - and then listening, speaking - and then listening. Maintaining a quivering trigger-finger poised over the Google button does not, in my highly-opinionated opinion, qualify a person for being a conversationalist. It rather seems more like a slavering urge to instantly debunk the other person's merest utterance. "Looks like rain." "Let's google that!" - poke, poke, button, button - "Nope! You're wrong! Sunny all day! HA! " Or maybe - "I think Apollo 11 landed in the summer of '68 - I can remember it was just before my birthday." "Let's google that!" - poke, poke, button, button - "Nope! You're wrong! '69! HA!" It's like some horrible parody of discourse - and that's not even to mention the addled scrabbling of all the text-your-girlfriend-while-you're-talking-to-your-mother little monsters out there who are poised to assume control of our civilization.
Ah, well - soon enough implants will resolve all this confusion. Thoughts will be duly sourced from appropriate algorithms. Conversation will evolve into a fidgety dance of racing thumbs, now divorced from the very index fingers to which they were so elegantly opposed for so many aeons. My kind will grow lonely, wistful - talking more and more to the cat. And when asked to comment on some electronically-confirmed bit of information - oh, I can see it now........... "Oh yeah? Well, Google this!"
Phil Burpee
November 26, 2011


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Posted: June 10, 2009/Under: Operating Systems/By: SY Tan
After the release of Firefox 3.5 Beta 4 in April, Mozilla today released the Firefox 3.5 Preview for those users who have downloaded Firefox 3.5 Beta 4. According to Mozilla, more than 800,000 users have downloaded Firefox 3.5 beta 4 and these users will be receiving an update to a "Firefox 3.5 Preview".
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