What is Minimum: Definition and 1000 Discussions

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire domain (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

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  1. T

    Minimum value of an inequality

    Homework Statement Homework EquationsThe Attempt at a Solution i just straight up applied am gm ##\frac {x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq 3(\sqrt[3]{\frac{1}{(y+z)(x+z)(x+y)}}) ## so the denominator is which i had to maximise ## x^2(y+z) + y^2(x+z) +z^2(x + y) + 2\\...
  2. K

    Maximum, minimum, and continuity

    Homework Statement Theorem 3 that i will give in the attempt at a solution talks about a closed interval, here it's open Homework Equations Continuity: $$\vert x-c \vert < \delta~\Rightarrow~\vert f(x)-f(c) \vert < \epsilon$$ $$\delta=\delta(c,\epsilon)$$ The Attempt at a Solution
  3. G

    The relation between the energy minimum and entropy maximum

    Hi. This is the problem 5.1-1 from the second edition of Callen's Thermodynamics. It says Formulate a proof that the energy minimum principle implies the entropy maximum principle. That is, show that if the entropy were not maximum at constant energy then the enrgy could not be minimum at...
  4. R

    Minimum Kinetic energy of the hydrogen atom

    Homework Statement A hydrogen atom collides with another hydrogen atom at rest. If the electrons in both atoms are in the ground state, what is the minimum kinetic energy of the hydrogen atom such that the hydrogen atom at rest will have its electron in the first excited state after collision...
  5. S

    Find the minimum of c(3a+4b) when a^2+b^2 +c^2=1.

    Homework Statement Given a,b,c real and a^2+b^2 +c^2=1, to find the minimum of c(3a+4b) Homework EquationsThe Attempt at a Solution No positive clue yet.
  6. Const@ntine

    Minimum Height of a Vertical Reflector

    Homework Statement Find the minimum height of a vertical flat reflector, in which a 1,78 m person can see its full reflection. Homework Equations M = h'/h The Attempt at a Solution I really don't know what to do here, and I feel as if I'm missing some crucial element. The chapter about...
  7. B

    MHB Find the coordinates and the nature of any turning points (maximum and minimum)

    So I've been trying to wrap my head around this one for several hours now and it just has me stumped. I'm begging, someone, anyone, walk me through it before I swear off numbers for life hahah (Part b) Imgur: The most awesome images on the Internet Thanks in advance for any help anyone can...
  8. L

    Solving for Minimum point Excel

    Homework Statement [/B] Hello, I trying to solve for the minimum value that the x and y components reach in plot (A). I know that you can perform this using IgorPro but I would like to know how to solve for it using Excel. The plot was constructed using the following data points: 110...
  9. C

    I Minimum deviation in prism spectroscope

    Suppose that I use a prism (vertex angle ##\alpha##) spectroscope to analyze a beam of visible light from a mercury lamp (different wavelenghts) and I want the determine the refraction index of the prism using the minmum deviation angles ##D_{min}##...
  10. Arman777

    Magnetic moment and Minimum energy

    In exam we have a question like this , I said its false cause ##U=-\vec μ⋅\vec B## , ##U=-μBcosθ## when ##cosθ=1## it is in the most stable point.But when ##cosθ=-1## its not, since.In both cases ##\vec μ## and ##\vec B## are parallel ?
  11. R

    The minimum KE required for muons to travel a given distance

    Homework Statement Muons of Kinetic energy 'E' are produced in collision with a target in a laboratory. The mass of a muon is ##106MeV/c^2## and its half life in it's rest frame is ## 1.4 * 10^{-6}## in its rest frame. what should be the minimum KE such that more than half of the muons...
  12. Const@ntine

    Comp Sci C++: Method of Minimum Squares used for System Solving

    Homework Statement Okay, this one is a bit big, and I'm attempting a translation, so I'm going to post a TL;DR version at the bottom just to be sure. Anyway, here goes: While doing an experiment, we write down the values of a physical/natural size y, for various values of a physical/natural...
  13. E

    Battery charging: Minimum voltage differential?

    I have a question that I believe requires knowledge on the graduate level, but I can't necessarily express it in terms more complicated than plain English. If this is not so, moderator, please adjust to your taste, and thank you! If it helps to know this, I'm an audio/video systems designer and...
  14. S

    Minimum # of ops to even out an array by....

    I'm trying to solve https://www.hackerrank.com/challenges/equal which is how to make an array's N elements equal through operations restricted to (1) adding 1 to N-1 elements, (2) adding 2 to N-1 elements or (3) Adding 5 to N-1 elements. I started out by writing some test cases that might be...
  15. CynicusRex

    Minimum of x+1/x (Perimeter Square < Perim equal area rects)

    Homework Statement Gelfand - Algebra p.115 problem 264: Prove that a square has the minimum perimeter of all rectangles having the same area. Hint. Use the result of the preceding problem. Homework Equations Preceding problem: Prove that a square has the maximum area of all rectangles having...
  16. N

    B Do neutron stars have a minimum volume?

    Do neutron stars have a minimum volume? Anything "in the way" of perhaps baseball sized neutron star? Or would something like that be an impossibility? How neat to have one in a laboratory...or not lol
  17. Mr Davis 97

    How to use AM-GM to find the minimum of a function

    Homework Statement Let ##\displaystyle f(x) = 3 \left( x + \frac{1}{x} \right)##. Use AM-GM to find the minimum value. Homework EquationsThe Attempt at a Solution I know how to find the minimum value using calculus: we take the derivative, set it to zero,then find the critical points. Then we...
  18. JoePhysics

    A Problems from Lev Landau's "Theoretical Minimum"

    It seems Lev Landau created an entry exam to test his students, and the exam was known to be ridiculously hard. To get an idea as to how hard the test really was, I've been scouring the Internet for problems Landau proposed... so far I've managed to find only four. Electrodynamics A dielectric...
  19. V

    Computing Minimum Spanning Tree in R

    Hi guys. Could you please give me more information regarding the necessary library and/or the code I should use to compute the Minimum Spanning Tree in R? Many thanks.
  20. M

    B Find the Minimum Value of Expression Involving Positive Real Numbers

    if ## a,b,c,d,e ## are positive real numbers, minimum value of (a+b+c+d+e)( \frac{1}{a} +\frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e} ) (A) 25 (B) 5 (C) 125 (D) cannot be determined My approach : expanding the expression , i get 5+a( \frac{1}{b} +\frac{1}{c} +\frac{1}{d} +\frac{1}{e}...
  21. H

    Single slit with minimum uncertainty

    Homework Statement A horizontal beam of laser light of wavelength 474 nm passes through a narrow slit that has width 5.80×10−2 mm . The intensity of the light is measured on a vertical screen that is 2.40 m from the slit. Use the result of part A to estimate the width of the central diffraction...
  22. A

    What is the Strategy for Evaluating Minimum and Maximum Values in Calculus?

    Homework Statement The problem is in the attached file. The part I need a little help with is part b. Homework Equations and attempt at a solution[/B] For part a, I got h(8) = 2, h'(6) = -2, and h''(4) = -2. For part c, I found that the integral from 0 to 5 is 7, so I multiplied 7 by 7 to...
  23. lfdahl

    MHB What is the maximum and minimum sum for $|a-b|+|b-c|+|c-a|$ with given relation?

    Find the maximum and minimum of the sum $|a-b|+|b-c|+|c-a|$, if the integer numbers $a,b$ and $c$ satisfy the following relation: \[ (a-b)^3+(b-c)^3+(c-a)^3 = 60 \]
  24. mamadou

    What Is the Minimum Height for a Ball to Complete a Loop in Physics?

    Homework Statement An empty ball , of mass m moment of inertia I = (2m.r²)/3, is rolling across the path shown below : there is friction fr from A to C . r is the radius of the ball , and R is the radius of the circular part within the path . what would be the minimal height h , so...
  25. A

    Minimum tangential speed for orbit

    Hello. Is there a formula describing the shape of the orbit and the size of the orbit given the tangential and perpendicular speeds? This would be based on the assumptions that the planets are point particles, everything is in a vacuum, that there are no other gravitational fields, and that one...
  26. smodak

    Relativity Susskind's Third book on the Theoretical Minimum Listed

    Amazon now lists Susskind's third book on the theoretical minimum series Special Relativity and Classical Field Theory https://www.amazon.com/dp/0465093345/?tag=pfamazon01-20
  27. M

    MHB Minimum of function under constraint

    Hey! :o We want to minimize the function $g(x_1, x_2)=2x_1+ x_2$ under the constraint $f(x_1, x_2)=x_1\cdot x_2=18$. \begin{equation*}x_1\cdot x_2=18 \Rightarrow x_1=\frac{18}{x_2}\end{equation*} \begin{equation*}\tilde{g}(x_2)=g\left (\frac{18}{x_2}, x_2\right )=2\cdot \frac{18}{x_2}+ x_2=...
  28. L

    The Minimum voltage requirement for an RC Oscillator

    Hi I am trying to run an RC oscillator using a 2N3904 (Datasheet for reference: http://www.kynix.com/uploadfiles/pdf8798/2N3904.pdf) The circuit looks like the following. I want to run it at 0-5v but I failed, I assume it is because the voltage is not high enough? What would you expect the...
  29. Samar A

    B What is the minimum voltage required to get characteristic spectrum

    Hi, The x-ray consists of the bremsstrahlung spectrum and the characteristic spectrum. We can get x-rays by using Coolidge tube where there are an applied voltage on the tube between the filament and the target. The bremsstrahlung is depeding on the PD between the filament and the target, but it...
  30. Bekamop99

    Minimum work required to put object into orbit?

    Homework Statement Scientists are planning to launch a rocket from the surface of the Earth into an orbit at a distance of 18000 km above the centre of the Earth. The radius of the Earth is 6400 km and it has mass 6.0x10^24 kg. What is the minimum work done to move the 150kg mass of the rocket...
  31. doktorwho

    What Is the Minimum Force Needed to Keep Two Blocks Stationary?

    Homework Statement ##m=1[kg]##, ##M=5[kg]##, ##u=0.6## Calculate the minimum force required for the blocks to remain in the same position thourought the motion. For the part a) there is no friction between block 2 and the floor, for the part b there is and its ##u_f## Homework Equations 3...
  32. Aleph_null

    Find the minimum and maximum of a ratio for equilibrium

    Homework Statement Consider the rod as shown in the figure. The coefficient of static friction at A and B is \mu_s. I'm trying to find how big and how small the ratio l/d can be for the rod to still be in equilibrium. I have calculated how big all forces are acting upon the rod in the case...
  33. J

    Minimum safe height for a roller coaster

    Homework Statement Based on the height of a first hill (114.5m), mathematically determine the minimum height (circular motion and radius) of the next hill based on the speed generated on the first hill. At the bottom of the first hill, the velocity of the roller coaster is 47.38m/s. There is...
  34. Albert1

    MHB Finding Minimum Value of $n$ for Given Sum and Product

    $n\in N,\,\,and \,\, a_1,a_2,a_3,-------,a_n\in Z$ $if \,\, a_+a_2+a_3+-----+a_n=a_1\times a_2\times a_3\times------\times a_n=2006$ $find \,\, min(n)$
  35. E

    Minimum coefficient of static friction

    Homework Statement If the incline of the driveway is 19 ∘ from the horizontal, what must the minimum coefficient of static friction be between your shoes and the ice? Homework Equations Fs=uN The Attempt at a Solution
  36. L

    What is the minimum mass of the barbell without tipping?

    Homework Statement Typical weight sets used for bodybuilding consist of disk-shaped weights with holes in the center that can slide onto 2.20 m long barbells. A barbell is supported by racks located a fifth of its length from each end, as shown in the figure. What is the minimum mass m of the...
  37. O

    Minimum radius of bubbles in a soda bottle (surface tension)

    Homework Statement [/B] The question is from chapter 9 of "Exercises in Introductory Physics" by Leighton and Vogt. The answer given in the book is ##R = 4.9 \times 10^{-5} \rm{cm}##. Homework Equations $$\sigma = \frac{\Delta P \cdot R}{4}$$ Where, ##\sigma## is the surface tension...
  38. P

    Finding the Minimum Work Done for a Thermodynamic Process

    Homework Statement Given two indistinguishable objects at the same initial temperature ##T_i##, calculate the minimum work done by a refrigerator functioning between the two objects till one of the objects reaches a new temperature ##T_2##, assume constant heat capacities and constant pressure...
  39. A

    CDF of minimum of N random variables.

    There's this problem that I've been trying to solve. I know the solution for it now but my initial attempt at a solution was wrong and I can't seem to figure out the mistake with my reasoning. I'd appreciate some help with figuring this one out. 1. Homework Statement I have a set of random...
  40. F

    A Minimum uncertainty in electron position

    I am wondering about the minimum possible uncertainty (standard deviation) in an electron's position (Δx). How precise can one know the electron's whereabouts without creating other sorts of particles and phenomenons. I know of the localization energy interpretation of the energy uncertainty...
  41. lfdahl

    MHB What is the minimum value of this summation with given constraints?

    Find the minimum of the sum: \[\sum_{i=1}^{5}x_i\], where $x_i \ge 0$ and $\sum_{i<j}|x_i-x_j| = 1.$
  42. A

    What is the minimum horizontal force?

    Homework Statement What is the minimum horizontal force F needed to make the box start moving in (Figure 1) ? The coefficients of kinetic and static friction between the box and the floor are 0.25 and 0.46, respectively. Homework Equations Fs= mu * N The Attempt at a Solution So I know that...
  43. A

    How to find the minimum horizontal force?

    Homework Statement What is the minimum horizontal force F needed to make the box start moving in (Figure 1) ? The coefficients of kinetic and static friction between the box and the floor are 0.28 and 0.39, respectively. Homework Equations fk=mu N fs=mu N The Attempt at a Solution Im not...
  44. moenste

    Minimum safe distance from a radioactive source

    Homework Statement The potassium isotope 4219K has a half-life of 12 hr, and disintegrates with the emission of a γ-ray to form the calcium isotope 4220Ca. What other radiation besides γ-rays must be emitted? How many electrons, protons, and neutrons are there in an atom of the calcium isotope...
  45. Daniel Engebretsen

    Glass fibre Aquarium pool minimum thickness?

    Hello, I am working on a Project and it has been a while since i did calculations With mechanics. The pool would be made in glassfibre with polyester and will be: L=2000mm W= 1750mm h= 400mm we use the E-module for glass, so we use E= 70 000 N/mm2 I= bh3/12 = 2000mm x 4003 mm /12 = 1,067 x...
  46. N

    Books for the theoretical minimum video courses

    Hi, everyone. I'm currently taking a gap year, and my plan is to study physics, probably for the rest of my life (With that I mean, reading and learning physics for the rest of my life ;-) ) But I thought why not start now? I kind of have fallen in love with the lectures held by Leonard...
  47. afcsimoes

    B Is there a minimum wavelength for electromagnetic radiation?

    The wavelength is inversely proportional to the foton energy. So, the limit can be stated by the mass of the full universe. But how much near that limit can the light be?
  48. G

    Minimum velocity required by grasshopper

    Homework Statement There is a grasshopper who wants to cross a log(cylindrical) of radius $$R$$.Calculate the minimum velocity required by grasshopper to just scrape the log. Homework Equations First of all I have a doubt.So equations are out of question. At least I know that it is a question...
  49. Amru123

    About which axis the moment of inertia of A body is minimum

    I remember my teacher saying it to be the axis along the centre of mass but the centre of mass has all the mass concentrated in it and hence moment of inertia should increase as moment of inertia is proportional to mass?
  50. Mr Davis 97

    Minimum time for rowing across a river and running

    Homework Statement In hot pursuit of a suspect, Agent Logan of the FBI must get directly across a 1200m wide river in minimum time. The river's current is 0.80m/s, she can row a boat at 1.60m/s and she can run 3.00m/s. Describe the path she should take (rowing + running along the shore) for the...
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