Category Archives: applications of maths

The Greatest Auction Ever Held

Reference: A Beautiful Mind by Sylvia Nasar.

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Washington, D.C., December 1994:

On the afternoon of December 5, 1994, John Nash was riding in a taxi headed to Newark Airport on his way to Stockholm, where he would, in a few days time, receive from the King of Sweden the gold medal engraved with the portrait of Alfred Nobel. At around the same time, a few hundred miles in the south, in downtown Washington, D.C., Vice President Al Gore was announcing with great fanfare the opening of the “greatest auction ever.”

There was, as The New York Times would later report, no fast-talking auctioneer, no banging gavel, no Old Masters. On the auction block was thin air — airwaves that could be used for the new wireless gadgets like telephones, pagers, faxes — worth billions and billions of dollars, enough licenses for every major American city to have three competing cellular phone services. In the secret war rooms and building booths were CEOs of the world’s biggest communications conglomerates — and unlikely group of blue sky economic theoreticians who were advising them.When the auction finally closed the following March, the winning bids totaled more than $7 billion making it the biggest sale in American history of public assets and one of the most successful (and lucrative) applications of economic theory to public policy ever. Michael Rothschild, dean of Princeton’s Woodrow Wilson School, later called it “a demonstration that people thinking hard about a problem can make the world work better…a triumph of pure thought.”

The juxtaposition of Gore and Nash, the high-tech auction and the medieval pomp of the Nobel ceremony, was hardly an accident. The FCC auction was designed by young economists who were using tools created by John Nash, John Harsanyi and Reinhard Selten. Their ideas were specifically designed for analyzing rivalry and cooperation among a small number of rational players with a mix of conflicting and similar interests: people, governments, and corporations — and even animal species.

The prize itself was a long overdue acknowledgement by the Nobel committee that a sea change in economics, one that had been underway for more than a decade, had taken place. As a discipline, economics had long been dominated by Adam Smith’s brilliant metophor of the Invisible Hand. Smith’s concept of perfect competition envisions so many buyers and sellers that no single buyer or seller has to worry about the reactions of others. It is a powerful idea, one that predicted how free-market economies would evolve and gave policy-makers a guide for encouraging growth and dividing the economic pie fairly. But in the world of mega-mergers, big government, massive foreign direct investment, and whole-sale privatization, where the game is played by a handful of players, each taking into account the others’ actions, each pursuing his own best strategies, game theory has come to the fore.

After decades of resistance —- Paul Samuelson used to joke about “the swamp of n-person game theory” —- a younger generation of theorists began using game theory in areas from trade to industrial organization to public finance in the late 1970’s and early 1980’s. Game theory opened up “terrain for systematic thinking that was previously closed.” Indeed, as game theory and information economics have become increasingly entwined, markets traditionally seen as fitting the purely competitive mold have increasingly been studied using game theory assumptions. The latest generation of texts used in top graduate schools today all recast the basic theories of the firm and the consumer, the foundation of economics, in terms of strategic games. “Concepts, terminology and models from game theory have come to dominate many areas of economics,” said Avinash Dixit, an economist at Princeton who uses game theory in work on international trade and is the author of Thinking Strategically. “At last we are seeing the realization of the true potential of the revolution launched by von Neumann and Morgenstern.” And because most economic applications of game theory use the Nash equilibrium concept, “Nash is the point of departure.”

The revolution has gone far beyond research journals, experimental laboratories at Caltech and the University of Pittsburgh, and classrooms of elite business schools and universities. The current generation of economic policy-makers — including Lawrence Summers, undersecretary of the treasury, Joseph Stiglitz, chairman of the Council of Economic Advisers, and Vice-President Al Gore — are steeped in the stuff, which they say, is useful for thinking about everything from budget proposals to Federal Reserve policy to pollution cleanups.

The most dramatic use of game theory is by governments from Australia to Mexico to sell scarce public resources to buyers best able to develop them. The radio spectrum, T-bills, oil leases, timber, and pollution rights are now sold in auctions designed by game theorists — with far greater success than that of earlier policies.

Economists like Nobel Laureate Ronald Coase have advocated the use of auctions by government since the 1950’s. Auctions have long been used in markets where sellers of unusual items — from vintage wines to movie rights — have no idea what bidders are willing to pay. Their basic purpose is to make bidders reveal how much they value the item. But the arguments of Coase and others were stated in abstract, entirely theoretical terms, and little thought was given to how such auctions would actually be conducted. Congress remained skeptical.

Before 1994, Washington simply gave away licenses for free. Until 1982, it had been up to regulators to decide which companies deserved the licenses. Needless to say, the process was dominated by political pressures, outrageously expensive paperwork, and long delays. The pace of licensing lagged hopelessly behind market shifts and new technologies. After 1982, Washington awarded licenses using lotteries, with the winners free to resell licenses. Although the reform did speed up the granting of licenses, the process was still hugely inefficient — and unfair. Bidders with no intention of operating an actual telephone business spent millions to get into the game for the purpose of reaping a windfall. Further, although telephone companies were forced to pay the costs of obtaining licenses, Washington (and taxpayers) did not get the benefits of any revenues. There had to be a better way.

A young generation of game theorists, including Paul Milgrom, John Roberts, and Robert Wilson at the Stanford B-school, came up with that better way. Their chief contribution consisted of recognizing, as Milgrom said, that “the mere design of some auction was not enough…Getting the auction design right was also critically important.” In particular, they concluded that the most obvious auction designs —- auctioning licenses one by one in sequence using simultaneous sealed bids — was the way least likely to succeed in getting licenses into the hands of corporations that could use them best —- Washington’s stated objective. Game theorists treat an auction like a game with rules and try to evaluate how a given set of rules, taken together, is apt to affect the bidders’ behaviour. They take stock of the options the rules allow, the payoffs to the bidders associated with the options, and bidders expectations about their competitors’ likely choices.

Why did these economists conclude that traditional auction formats would not work? Mainly because the value of each individual license to a user depends — as is the case with a Rembrandt or a Picasso — on what other licenses the user is able to obtain. Some licenses are perfect substitutes for one another. That would be the case for similar spectrum bands to provide a given service. But others are complements. That would be the case for licenses to provide paging services in different parts of the country.

“To permit the efficient license assignment, an auction must allow bidders to consider various packages of licenses, combining complements and switching among substitutes during the course of the auction. Designing an auction to allow this is quite difficult,” writes Paul Milgrom., one of the economists who designed the FCC auction of which Gore was speaking.

A second source of complexity, Milgrom says, is that the purpose of the licenses is to create businesses for new services with unknown technology and unknown consumer demand. Since bidders’ opinions are bound to be wildly divergent, it is possible that license assignment would depend more on bidders’ optimism than on their ability to create a desired service. Ideally, an auction design can minimize that problem.

As Congress and the FCC inched closer to the notion of auctioning off spectrum rights, Australia and New Zealand both conducted spectrum auctions. That they proved to be costly flops and political disasters illustrated that the devil really was in the details. In New Zealand, the government ran a so-called second price auction, and newspapers were full of stories about winners who paid far below their bids. In once case, the high bid was NZ$7 million, the second bid was NZ$5000, and the winner paid the lower price. In another, an Otago University student bid NZ$1 for a television license in a small city. Nobody else bid, so he got it for one dollar. The government expected the cellular licenses to fetch NZ$240 million. The actual revenue was NZ$36 million, one-seventh of the advance estimate. In Australia, a botched auction, in which parvenu bidders pulled the wool over the government’s eyes, delayed the introduction of pay television by almost a year.

The FCC’s chief economist was an advocate of auctions, but no game theorists were involved in the first stage of the FCC auction design. The theorists’s phones started ringing only by accident after the FCC issued a tentative proposal for an auction format with dozens of footnotes to the theoretical literature on auctions. That was how Milgrom and his colleague Robert Wilson, leading auction theorists, got into the game. Milgrom and Wilson proposed that the FCC adopt a simultaneous, multiple round auction. In a simultaneous auction, a bunch of licenses are sold at the same time. Multiple rounds means that, after the first round of bidding prices are announced, and bidders have a chance to withdraw or raise one another’s bids. This is repeated round after round until the auction is over. The chief advantage of this format is that it allows bidders to take account of interdependencies among licenses. Just as sequential, closed-bid auctions let sellers discover what bidders are willing to pay for individual items, the simultaneous, ascending-bid auction lets them discover the market value of different groupings of items.

This early proposal —- which the FCC eventually adopted — did not cover seemingly small but critical details. Should there be deposits? Minimum bid increments? Time limits? Should the bidding system be wholly computerized or executed by hand? And so forth. Milgrom, Roberts and another game theorist, Preston McAfee, an adviser to AirTouch, provided proposals on these issues. The FCC hired another game theorist, John McMillan, of the University of Caliofornia, San Diego, to help evaluate the effect of every proposed rule. According to Milgrom, “Game theory played a central role in the analysis of the rules. Ideas of Nash equilibrium, rationalizability, backward induction, and incomplete information, though rarely named explicitly, were the real basis of daily decisions about the details of the auction process.

By late spring 1995, Washington had raised more than USD 10 billion from spectrum auction. The press and the politicians were ecstatic. Corporate bidders were largely able to protect themselves from predatory bidding and were able to assemble an economically sensible set of licenses. It was, as John McMillan said, ” a triumph for game theory.”

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PS: a triumph of pure mathematical thought !:-)

Regards,

Nalin Pithwa

 

 

 

 

 

The importance of lines and slopes

  1. Light travels along straight lines. In fact, the shortest distance between any two points is the path taken by a light wave to travel from the initial point to the final point. In other words, it is a straight line. (A slight detour: using this elementary fact, can you prove the triangle inequality?)
  2. Bodies falling from rest in a planet’s gravitational field do so in a straight line.
  3. Bodies coasting under their own momentum (like a hockey puck gliding across the ice) do so in a straight line. (Think of Newton’s First Law of Motion).

So we often use the equations of lines (called linear equations) to study such motions.

Many important quantities are related by linear equations. Once we know that a relationship between two variables is linear, we can find it from any two pairs of corresponding values just as we find the equation of a line from the coordinates of two points.

Slope is important because it gives us a way to say how steep something is (roadbeds, roofs, stairs, banking of railway tracks). The notion of a slope also enables us to describe how rapidly things are changing. (To philosophize, everything in the observable universe is changing). For this reason, slope plays an important role in calculus.

More later,

Nalin Pithwa.

PS: Ref: Calculus and Analytic Geometry by G B Thomas and Finney; or any other book on calculus.

PS: I strongly recommend the Thomas and Finney book : You can get it from Amazon India or Flipkart:

https://www.amazon.in/Thomas-Calculus-George-B/dp/9353060419/ref=sr_1_1?crid=36S3685TG7OYF&keywords=thomas+calculus&qid=1561503390&s=books&sprefix=Thomas+%2Caps%2C259&sr=1-1

or Flipkart:

https://www.flipkart.com/thomas-calculus-1/p/itmebug5kzrnttfj?pid=9789332547278&lid=LSTBOK9789332547278CHN4GH&marketplace=FLIPKART&srno=s_1_23&otracker=AS_Query_OrganicAutoSuggest_2_9&otracker1=AS_Query_OrganicAutoSuggest_2_9&fm=SEARCH&iid=fdc8327b-756c-4f6d-aa10-b45acc900e12.9789332547278.SEARCH&ppt=sp&ppn=sp&ssid=uz7zckp71c0000001561503474614&qH=2488f76736a10369

101 Careers in Mathematics: Andrew Sterrett, MAA publication

https://www.maa.org/press/maa-reviews/101-careers-in-mathematics

Shared by Nalin Pithwa — for spreading awareness in India also about career opportunities in maths/mathematics

 

 

More questions on applications of derivatives: IITJEE mains maths tutorial

  1. Prove that the minimum value of (a+x)(b+x)/(c+x) for x>-c, is (\sqrt{a-c}+\sqrt{b-c})^{2}.
  2. A cylindrical vessel of volume 25\frac{1}{7} cubic meters, open at the top, is to be manufactured from a sheet of metal. Find the dimensions of the vessel so that the amount of metal used is the least possible.
  3. Assuming that the petrol burnt in driving a motor boat varies as the cube of its velocity, show that the most economical speed when going against a current of c kmph is 3c/2 kmph.
  4. Determine the altitude of a cone with the greatest possible volume inscribed in a sphere of radius R.
  5. Find the sides of a rectangle with the greatest possible perimeter inscribed in a semicircle of radius R.
  6. Prove that in an ellipse, the distance between the centre and any normal does not exceed the difference between the semi-axes.
  7. Determine the altitude of a cylinder of the greatest possible volume which can be inscribed in a sphere of radius R.
  8. Break up the number 8 into two parts such that the sum of their cubes is the least possible. (Use calculus techniques, not intelligent guessing).
  9. Find the greatest and least possible values of the following functions on the given interval: (i) y=x+2\sqrt{x} on [0,4]. (ii) y=\sqrt{100-x^{2}} on [-6,8] (iii) y=\frac{a^{2}}{x}+\frac{b^{2}}{1-x} on (0,1) with a>0 and b>0 (iv) y=2\tan{x}-\tan^{2}{x} on [0,\pi/2) (iv) y=\arctan{\frac{1-x}{1+x}} on [0,1].
  10. Prove the following inequalities: (in these subset of questions, you can try to use pure algebraic methods, apart from calculus techniques to derive alternate solutions): (i) 2\sqrt{x} >3-\frac{1}{x} for x>1 (ii) 2x\arctan{x} \geq \log{(1+x^{2})} (iii) \sin{x} < x-\frac{x^{3}}{6}+\frac{x^{5}}{120} for x>0. (iv) \log{(1+x)}>\frac{\arctan{x}}{1+x} for x>0 (iv) e^{x}+e^{-x} > 2+x^{2} for x \neq 0.
  11. Find the interval of monotonicity of the following functions: (i) y=x-e^{x} (ii) y=\log{(x+\sqrt{1+x^{2}})} (iii) y=x\sqrt{ax-x^{2}} (iv) y=\frac{10}{4x^{3}-9x^{2}+6x}
  12. Prove that if 0<x_{1}<x_{2}<\frac{\pi}{2}, then \frac{\tan{x_{2}}}{\tan{x_{1}}} > \frac{x_{2}}{x_{1}}
  13. On the graph of the function y=\frac{3}{\sqrt{2}}x\log{x} where x \in [e^{-1.5}, \infty), find the point M(x,y) such that the segment of the tangent to the graph of the function at the point intercepted between the point M and the y-axis, is the shortest.
  14. Prove that for 0 \leq p \leq 1 and for any positive a and b, the inequality (a+b)^{p} \leq a^{p}+b^{p} is valid.
  15. Given that f^{'}(x)>g^{'}(x) for all real x, and f(0)=g(0), prove that f(x)>g(x) for all x \in (0,\infty), and that f(x)<g(x) for all x \in (-\infty, 0).
  16. If f^{''}(x)<0 for all x \in (a,b), prove that f^{'}(x)=0 at most once in (a,b).
  17. Suppose that a function f has a continuous second derivative, f(0)=0, f^{'}(0)=0, f^{''}(x)<1 for all x. Show that |f(x)|<(1/2)x^{2} for all x.
  18. Show that x=\cos{x} has exactly one root in [0,\frac{\pi}{2}].
  19. Find a polynomial P(x) such that P^{'}(x)-3P(x)=4-5x+3x^{2}. Prove that there is only one solution.
  20. Find a function, if possible whose domain is [-3,3], f(-3)=f(3)=0, f(x) \neq 0 for all x \in (-3,3), f^{'}(-1)=f^{'}(1)=0, f^{'}(x)>0 if |x|>1 and f^{'}(x)<0, if |x|<1.
  21. Suppose that f is a continuous function on its domain [a,b] and f(a)=f(b). Prove that f has at least one critical point in (a,b).
  22. A right circular cone is inscribed in a sphere of radius R. Find the dimensions of the cone, if its volume is to be maximum.
  23. Estimate the change in volume of right circular cylinder of radius R and height h when the radius changes from r_{0} to r_{0}+dr and the height does not change.
  24. For what values of a, m and b does the function: f(x)=3, when x=0; f(x)=-x^{2}+3x+a, when 0<x<1; and f(x)=mx+b, when 1 \leq x \leq 2 satisfy the hypothesis of the Lagrange’s Mean Value Theorem.
  25. Let f be differentiable for all x, and suppose that f(1)=1, and that f^{'}<0 on (-\infty, 1) and that f^{'}>0 on (1,\infty). Show that f(x) \geq 1 for all x.
  26. If b, c and d are constants, for what value of b will the curve y=x^{3}+bx^{2}+cx+d have a point of inflection at x=1?
  27. Let f(x)=1+4x-x^{2} \forall x \in \Re and g(x)= \left\{  \begin{array}{ll} max \{ f(x): x\leq t\leq x+3\} & 0 \leq x \leq 3\\ min (x+3) & 3 \leq x \leq 5 \end{array} \right. Find the critical points of g on [0,5]
  28. Find a point P on the curve x^{2}+4y^{2}-4=0 so that the area of the triangle formed by the tangent at P and the co-ordinate axes is minimum.
  29. Let f(x) = \left \{ \begin{array}{ll} -x^{3}+\frac{b^{3}-b^{2}+b-1}{b^{2}+3b+2} & 0 \leq x <1 \\ 2x-3 & 1 \leq x \leq 3 \end{array}\right. Find all possible real values of b such that f(x) has the smallest value at x=1.
  30. The circle x^{2}+y^{2}=1 cuts the x-axis at P and Q. Another circle with centre Q and variable radius intersects the first circle at R above the x-axis and line segment PQ at x. Find the maximum area of the triangle PQR.
  31. A straight line L with negative slope passes through the point (8,2) and cuts the positive co-ordinate axes at points P and Q. Find the absolute minimum value of OP+PQ, as L varies, where O is the origin.
  32. Determine the points of maxima and minima of the function f(x)=(1/8)\log{x}-bx+x^{2}, with x>0, where b \geq 0 is a constant.
  33. Let (h,k) be a fixed point, where h>0, k>0. A straight line passing through this point cuts the positive direction of the co-ordinate axes at points P and Q. Find the minimum area of the triangle OPQ, O being the origin.
  34. Let -1 \leq p \leq 1. Show that the equations 4x^{3}-3x-p=0 has a unique root in the interval [1/2,1] and identify it.
  35. Show that the following functions have at least one zero in the given interval: (i) f(x)=x^{4}+3x+1, with [-2,-1] (ii) f(x)=x^{3}+\frac{4}{x^{2}}+7 with (-\infty,0) (iii) r(\theta)=\theta + \sin^{2}({\theta}/3)-8, with (-\infty, \infty)
  36. Show that all points of the curve y^{2}=4a(x+\sin{(x+a)}) at which the tangent is parallel to axis of x lie on a parabola.
  37. Show that the function f defined by f(x)=|x|^{m}|x-1|^{n}, with x \in \Re has a maximum value \frac{m^{m}n^{n}}{(m+n)^{m+n}} with m,n >0.
  38. Show that the function f defined by f(x)=\sin^{m}(x)\sin(mx)+\cos^{m}(x)\cos(mx) with x \in \Re has a minimum value at x=\pi/4 which m=2 and a maximum at x=\pi/4 when m=4,6.
  39. If f^{''}(x)>0 for all x \in \Re, then show that f(\frac{x_{1}+x_{2}}{2}) \leq (1/2)[f(x_{1})+f(x_{2})] for all x_{1}, x_{2}.
  40. Prove that (e^{x}-1)>(1+x)\log(1+x), if x \in (0,\infty).

Happy problem solving ! Practice makes man perfect.

Cheers,

Nalin Pithwa.

 

 

 

Maths and the Bomb: Sir Michael Atiyah at 80

Just paying yet another tribute to Sir Michael Atiyah (re-sharing one of the articles I have collected about him):

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Maths and the bomb Sir Michael Atiyah at 80

The Times

April 21 2009

When, five years ago, he shared the £480,000 Abel Prize, the equivalent of a Nobel prize in the world of mathematics, Sir Michael Atiyah might have listened to his wife’s urgings to put his feet up and settle into a comfortable life. But that would not have been his style. “Some mathematicians retire,” he concedes with a smile. “I don’t think I have.”

This week, Sir Michael’s 80th birthday and a life dedicated to science and political activism is celebrated in a series of events. A three-day conference celebrating his contribution to geometry and physics, at the University of Edinburgh Informatics Forum, ends today, his birthday. Tomorrow and on Friday his Sir Michael’s role in promoting disarmament is recognised with readings and lectures dedicated to exposing the folly of nuclear weapons.

Much has been achieved at an age when contemporaries might have settled for a quiet life. In 1995, as president of the Royal Society and aged 67, Sir Michael made a stinging attack on Britain’s nuclear weapons policy.

Subsequently he accepted the presidency of the influential Pugwash disarmament conferences, which unite scientists in opposition to the arms race.

He still believes passionately in the cause, which, he says, is more important to the world than maths, “because if we blow ourselves up, there will be no mathematics anyway”.

Sir Michael discovered his aptitude for mathematics during his boyhood in the Sudan. His Lebanese father was an Oxford graduate and a civil servant, his mother was Scottish and he grew up regarding himself as British, studying at Manchester Grammar School and Cambridge University.

The key professional encounters in his life came in the United States in the 1950s, when he joined the Institute for Advanced Study, at Princeton University, a gathering place for the world’s most brilliant mathematical minds. Here he forged relationships which have endured, and much of his greatest work has come from what he calls the “dialogues of ideas” established there.

His greatest achievement has been the Atiyah-Singer theorem, which secured his fame and prize money, shared with his collaborator, Isadore Singer, of the US. At the time, he said he couldn’t think what to do with his share; the sporty red Lexus parked outside the Informatics building suggests he has since given it more thought.

In simple terms, the theorem provided a kind of analytical bridge which could be shifted between disciplines. “The theorem technique enables you to get to an answer by-passing all the intervening calculations,” he says. The idea “was something where you could calculate numbers of solutions by very indirect methods which applied in a very wide range of situations: geometry, algebra, physics…”

Maths, he says, is something he plays out in his mind as he walks around his flat and his garden, and he jots things down – “the dull stuff” – only when he has to check something.

“Walking helps the physiological process. You have to maintain a very high pitch of concentration when you do mathematics. It’s illumination – shining the mind’s eye on a problem and really seeing through it.

“The old clichés about the beauty of maths are true. It has beauty within it, but not all parts are equally beautiful. Beauty in mathematics is the thing that helps you in the search for truth.”

Some people, he believes, are born with mathematical brains, although they might choose other careers. One former student won the Nobel Prize for Economics, another is the best-paid hedge fund manager in the US. So was Sir Michael never tempted to use his mathematical skill in a wider world? Could he have solved the global financial crisis?

“Economics is a combination of gambling, psychology and who knows what,” he says. “The current crisis? I think people made a bloody mess. You can foretell that the bubble will burst – the question is when. If you gambled on it you might win or lose a lot of money. I just didn’t gamble.”

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Regards,

Nalin Pithwa.

 

Sir Michael Atiyah passes away; tribute by The New York Times

Permutations and Combinations: A primer only

Math moments: uses of mathematics in today’s world

Mathematics versus Physics

The object of pure Physics is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence. — J. J. Sylvester.

In my opinion, for example, Boole’s Laws of (Human) Thought. 

Applications of Derivatives: Training for IITJEE Maths: Part V

Question I:

Show that the equation of the tangent to the curve x=a \frac{f(t)}{h(t)} and y=a \frac{g(t)}{h(t)} can be represented in the form:

\left| \begin{array}{ccc} x & y & a \\ f(t) & g(t) & h(t) \\ f^{'}(t) & g^{'}(t) & h^{'}(t) \end{array} \right|=0

Question 2:

Show that the derivative of the function f(x) = x \sin{\frac{\pi}{x}}, when x>0 and f(x)=0, when x=0 vanishes on an infinite set of points of the interval $latex (0,1), Hint: Use Rolle’s theorem.

Question 3:

Prove that \frac{1}{1+x}<\log (1+x) < x for x>0. Use Lagrange’s theorem.

Question 4:

Find the largest term in the sequence a_{n}=\frac{n^{2}}{n^{3}+200}. Hint: Consider the function f(x)=\frac{x^{2}}{x^{3}+200} in the interval [1,\infty).

Question 5:

A point P is given on the circumference of a circle with radius r. Chords QR are drawn parallel to the tangent at P. Determine the maximum possible area of the triangle PQR.

Question 6:

Find the polynomial f(x) of degree 6, which satisfies \lim_{x \rightarrow 0} (1+\frac{f(x)}{x^{3}})=e^{2} and has a local maximum at x=1 and local minimum at x=0 and 2.

Question 7:

For the circle x^{2}+y^{2}=r^{2}, find the value of r for which the area enclosed by the tangents drawn from the point (6,8) to the circle and the chord of contact is maximum.

Question 8:

Suppose that f has a continuous derivative for all values of x and f(0)=0, with |f^{'}(x)| <1 for all x. Prove that |f(x)| \leq |x|.

Question 9:

Show that (e^{x}-1)>(1+x)\log {1+x}, if x \in (0,\infty).

Question 10:

Let -1 \leq p \leq 1. Show that the equations 4x^{3}-3x-p=0 has a unique root in the interval [1/2,1] and identify it.