(Calculus) Derivative Thinking Question

Recently, my Calculus and Vectors (Grade 12) teacher gave our class a thinking question/assignment to work on over the march break, and after working on for some time, I’ve become stuck on it.

The Question:

Consider f(x), a general quadratic function in standard form, and g(x) its reciprocal. For which values of x are the slopes of their respective tangent lines equal?

Consider two cases: one where it is true for exactly one value of x, and the other where it is true for exactly two values of x. In the latter case, you can assume that the steepness a does not equal 0 of f(x) is equal to both it’s y-intercept and also to the slope of its tangent at x = 1.

Find the required conditions on the parameters a,b,c in terms of a.

My Progress So Far:

So I know that f(x) = ax^2 +bx + c and g(x) = 1/f(x). After I solved for the derivative of each function, I set them both equal to each other and started solving for it. But then I came to a equation of (ax^2 + bx + c)^2 = -1 and that doesn’t work.

Next I tried something else. Since I know from the 2nd case that f(x) = c and f(x) = f'(1) when a cannot equal 0, I set c = f'(1) and got c = 2a + b. But after that, I don’t know where to go.

I’m not expecting a full solution, but if anyone could give me a hint for solving this question, I would appreciate it. I know that it says to write everything in terms of a, but I’m not sure how to approach that method.

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Thinking of boolean variables as sets, and $\mathbf{P=NP=co-NP}$?

This is an algorithm I came up with, that seeks to solve the ‘Boolean Tautology Problem’ in polynomial time, using $ 3-DNF$ clauses. I am posting this algorithm here seeking advice, and correction, if any can be provided.

I treat boolean variables as sets. The set itself denotes the variable to be $ true$ , and complement of the set denotes the variable to be $ false$ . $ \bigcap$ of sets denotes $ \land(AND)$ of the variables, and $ \bigcup$ of sets denotes $ \lor(OR)$ of the variables.

The sets are arranged so that they perfectly intersect, and every section is a boolean expression. Here are some examples(using upto $ 4$ sets/variables, since drawing more clutters the picture) :

$ 3$ variable example :

enter image description here

$ 4$ variable example :

enter image description here

Now, each clause in $ 3-DNF$ has $ 3$ literals(using the exact $ 3-DNF$ version). So, there are only $ 4$ possible cases for the literals :
1) All $ 3$ literals are positive – $ (a\land b\land c)$
2) $ 1$ literal negative, rest positive – $ (\lnot i\land j\land k)$
3) $ 2$ literals negative, remaining positive – $ (\lnot p\land\lnot q\land r)$
4) All $ 3$ literals negative – $ (\lnot x\land\lnot y\land\lnot z)$

Any $ 3-DNF$ expression consists of $ \lor$ of these $ 4$ kinds of clauses, and each clause itself is a subset of our total set of variables. Each clause highlights a certain part of the Venn Diagram of the set of variables. So, a tautological expression will have clauses that highlight the entirety of the Venn Diagram. The important part is that using $ 3-DNF$ , the total set can be highlighted only using these specific sets, and not just any arbitrarily small subset.

The way the algorithm works is by highlighting all such clauses found within the expression, and their intersections with all such other subsets. The reasoning is that since only the $ 4$ clause cases can highlight their respective subsets in the Venn Diagram, upon highlighting one such clause, other clauses which may intersect with it also have portions that need to be highlighted. Since the possible highlighter subsets are limited to the $ 4$ kinds of clauses, we just need to check intersections between them, two at a time.

Implementation :
The implementation would be as follows(given a $ 3-DNF$ expression with $ n$ variables) :
1) Create a graph, starting with all possible clauses as beginning nodes.(total number of clauses $ =8\times {n\choose 3}$ )
2) Create a set of children nodes for each parent node, by adding one more literal each to the parent clause.
3) For each such child, there is a dual child which has the negation of that literal.
4) Using these children as parents, follow procedure (2).
5) Continue this process until you have exhausted your variables, or have $ 6$ literals in your last created children nodes (intersection between two $ 3-DNF$ clauses).
6) A child can have multiple parents.

An example case of graph building is shown below. The directed arrow represents parent-to-child relation. Dual children are attached to the same relation for ease of view. The graph shown is partially built, so as to not clutter.

enter image description here

Solving :
Now, on to the solving the $ 3-DNF$ expression :
1) Scan through the expression and take up each clause.
2) For each clause, mark the corresponding node as true.
3) If a parent node is true, all child nodes are marked true.
4) If both dual children are true, the parent is marked true.
5) If every node is marked true, the expression is a tautology, otherwise not.

Remarks :
Step (3) of the solver ensures that if a particular clause is true, then all possible intersections of it, and any other clause also gets marked true. Since the graph stops building child nodes once those nodes have $ 6$ literals, we deal only with cases of intersection of two clauses. E.g.
$ (a\land b\land c)\land (\lnot d\land e\land\lnot f)=(a\land b\land c\land\lnot d\land e\land\lnot f)$
It should be obvious that if common literals exist in these clauses then it results in less than $ 6$ literals in the child clause. We do not need to check for intersections of more than two clauses, since the intersection will be contained within the intersection of two clauses each, since each true assignment is in disjunction with others.

Time Complexity :
Since we stop at child nodes with $ 6$ literals each, no matter the value of $ n$ , the number of variables, it is obviously polynomially bounded.

Closing words :
Please read through the method, and do tell me if there are any errors, or logical leaps that I made, or if the algorithm does actually work, which would be great.

EDIT : Added some subtitles for paragraphs.

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Atox builds a new ecosystem with a win-win thinking and returns to the initial wish of blockchain’s value

2019 is the year of return of blockchain’s value. Atox replaces the “family” thinking of “cutting leek” with “co-construction and sharing”, practices the original aim of the blockchain, and achieves a win-win new ecosystem and becomes an industry leader. 1.AtoX builds ecosystem with win-win thinking and becomes the leader of new mode of ecological construction

Ecological construction is an indispensable part of the blockchain project. Bitcoin, Ethereum, Ripple, EOS and other relatively leading blockchain projects, each has a strong ecosystem. However, most of the past projects used the idea of “cutting leeks” to engage in ecosystem, and the great technological invention of ICO was reduced from a credit machine into a fraud tool. As a result, the market ruthlessly fell, the lesson was very painful, and the blockchain revolution was almost ruined.

AtoX follows the spirit of mutual benefit of co-governance and sharing, and regards ecological construction as the strategic fulcrum of the blockchain project, making full use of its advantages, such as core technology, business model, model of token economy , ecological construction, team strength, capital and resource capabilities. To create a fast profitable, sustainable development, social contribution to achieve a unified blockchain project. The head effect has emerged and it has become a leader.

AtoX is the core value of the AtoX project culture by taking the ecosystem as the family and taking the eco-builders as family members. AtoX practices the eradication of ICO’s cutting leek, returning to the value source of the blockchain, allowing the ecosystem to grow together with the project, to get rich together, and to create a new ecological construction model.

Two new slogans from AtoX: not to be a leek but a community member, do not privately mining a mine but as a node. In the world of AtoX, the relation of production is new, and we work for ourselves. The core development teams, foundations, all-nodes, investors, commune members, and users will all participate on an equal footing. Everyone work together, actively contribute, jointly create the ecosystem, and share the ecological results. The AtoX ecosystem has its own token AXC to motivate all parties. The ecological profit comes from the return of the AtoX project to economic and social transformation services. Due to the huge energy of AtoX, the potential is unlimited.

2.AtoX has the strength to become the leader of win-win new ecological model

(1) Ai enhanced cross-chain technology

AtoX’s Ai enhances cross-chain, technically overcomes the triangular contradiction of decentralization, security and commerciality, and is a leading technical architecture. Currently, the AtoX ecosystem includes its own publci blockchain, its own wallet and exchange (its app is AtoXSwap) and its own token (AtoXCoin, abbreviated AXC).

AtoX’s Ai enhanced cross-chain architecture, which is very scalable and can do a lot of things. The lightning network achieves high-speed TPS, and the decentralization and security can be handed over to AI, which is the underlying public blockchain. In the new financial business model, such as payment, there are more places to use big data and Ai. Combine with business and users, and then gradually upgrade and improve after the business has been done.

(2) New financial business model

AtoX has pioneered the lightning network and payment settlement that is not affected by the bear market. It has built a channel to support dual-token UT/ST and to provide funds for entities to become the leading new financial platform to support blockchain reform.

(3) Sustainable Token Economy Model

AtoX wallets support cross-chain, also support multiple coins, and match sellers and buyers. AXC can be used as an intermediary to mine, and the price of AXC can be supported for a long time. Super nodes, alternate nodes, lightning nodes, each node generates new blocks, they can receive different rewards.

(4) International team

The AtoX project team is headquartered in Switzerland, with a core technical team based in Silicon Valley, USA, and there are also offices in Italy, Germany, Australia and other countries.

Strict compliance with Swiss rules is a guarantee for the interests of all parties involved in the project.

  1. Lightning network and payment settlement are not affected by fluctuations in coin prices, and they stand out and develop rapidly.

The AtoX project started with lightning network and payment settlement, and grasped the pulse of market development. It is a touch of color in the bear market. 2019 will be a year of rapid development of the lightning network. Lightning Network is a bitcoin-based micropayment network. The cryptocurrency can be effectively transferred across borders, and you can quickly move funds to where it is needed, do not involve geopolitics, without embargoes, sanctions or controlling traditional currencies. In the long run, cryptocurrencies can do what current payment systems, such as Visa, can’t do, that is, complete cross-border settlements in milliseconds and complete payments at very low commissions. The amount of payment can be not only less than 1 US dollar, but less than 1 cent, and the transfer time is less than one millisecond. With such cross-border settlement, a real innovative applications will emerge. Payment settlement will be a booming market. It is used to the payment and settlement of cryptocurrency and is not affected by currency fluctuations. Because when the value of cryptocurrency is low, it is not really affected by price fluctuations. For example, in the context of bear market in 2018, this market is also developing. If the system grows in flow and popularity, the volatility will naturally become smaller. The more people use it, the more they use it in their daily lives, the less volatility they have, because speculation is not the main driver. . 4. Who actively participate in AtoX will eventually become a rich harvester

From the whole market, the fall in 2018 is only the market, and the blockchain is still growing rapidly. For example, the number of wallet users in the Bitcoin blockchain has increased by 10 million from the beginning of 2017 to more than 31.9 million. China’s blockchain-related enterprises, According to the data of the State Administration for Industry and Commerce, there were 19,000 companies by the end of 2018, 5,000 at the end of last year, and nearly 15,000 registered in 2018.

In 2019, we believe that it is the year of making order out of confusion, value return, and not forgetting the original ideals and aspirations. The value of the blockchain’s credit mechanism, incentive mechanism, and organizational mechanism will once again become dominant. The top ten killer applications, such as new finance, digital asset trading, digital legal affairs, blockchain merchants, product traceability, distributed ledger, sharing economy, relief, new wisdom city, supervision sandbox and others will be born, leading the industry back to the bull market.

How do participants participate? If you have the technology to participate in the development, if you have the funds to invest, if you have the computing power to be a node, you don’t have anything, as long as you have passion, you can be a member of the ecological construction community.

AtoX is an ideal project to participate in. The standard is very simple. First, the project is promising, and second, the team is reliable.

We are believers, we believe that the logic of the blockchain transforming the world is right; we are devotees, we actively participate in project investment, business, activities; of course, we are qualified to become a harvester, and we can certainly become a harvester! Choose the AtoX project to let you break out in 2019. Participation link: Atox Project official website: [link removed] Atox project Chinese community: [link removed] Atox project English community: [link removed]

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