## 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, 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) 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; 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.

## Prof Ali Nesin, Eeelavati Prize 2018, and his Nesin Mathematics Village in Turkey

The only way I can pay an honour to an outstanding generous mathematician, Prof Ali Nesin, is to share information about him and his Mathematics Village is to share information collected from the internet:

http://www.nesinkoyleri.org/eng/

https://www.ams.org/notices/201506/rnoti-p652.pdf

https://en.wikipedia.org/wiki/Nesin_Mathematics_Village

https://interestingengineering.com/nesin-mathematics-village-learn-enjoy-math

https://www.middleeasteye.net/in-depth/features/turkey-s-mathematics-village-changing-education-one-equation-at-a-time-1597523620

http://mathematics-in-europe.eu/?p=1568

http://www.hurriyetdailynews.com/a-brief-introduction-to-turkeys-mathematics-village-70193

I can’t help myself noticing similarities between Prof Anand Kumar of India’s Super 30  http://www.super30.org    and Prof Ali Nesin 🙂

Regards,

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.

## References for IITJEE Foundation Mathematics and Pre-RMO (Homi Bhabha Foundation/TIFR)

1. Algebra for Beginners (with Numerous Examples): Isaac Todhunter (classic text): Amazon India link: https://www.amazon.in/Algebra-Beginners-Isaac-Todhunter/dp/1357345259/ref=sr_1_2?s=books&ie=UTF8&qid=1547448200&sr=1-2&keywords=algebra+for+beginners+todhunter
2. Algebra for Beginners (including easy graphs): Metric Edition: Hall and Knight Amazon India link: https://www.amazon.in/s/ref=nb_sb_noss?url=search-alias%3Dstripbooks&field-keywords=algebra+for+beginners+hall+and+knight
3. Elementary Algebra for School: Metric Edition: https://www.amazon.in/Elementary-Algebra-School-H-Hall/dp/8185386854/ref=sr_1_5?s=books&ie=UTF8&qid=1547448497&sr=1-5&keywords=elementary+algebra+for+schools
4. Higher Algebra: Hall and Knight: Amazon India link: https://www.amazon.in/Higher-Algebra-Knight-ORIGINAL-MASPTERPIECE/dp/9385966677/ref=sr_1_6?s=books&ie=UTF8&qid=1547448392&sr=1-6&keywords=algebra+for+beginners+hall+and+knight
5. Plane Trigonometry: Part I: S L Loney: https://www.amazon.in/Plane-Trigonometry-Part-1-S-L-Loney/dp/938592348X/ref=sr_1_16?s=books&ie=UTF8&qid=1547448802&sr=1-16&keywords=plane+trigonometry+part+1+by+s.l.+loney

The above references are a must. Best time to start is from standard VII or standard VIII.

-Nalin Pithwa.

## Arithmetic Progressions: A Primer for mathematical olympiads and IITJEE mathematics

a gentle introduction…it reminds us that math is about detecting and discerning patterns…whether in numbers, symbols, shapes/figures, or even logical reasoning…regards, Nalin Pithwa.