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.

View More On Wikipedia.org
Recent contents Top users
  1. R

    How to calculate the minimum torque required for a motor?

    Hi i want to choose a motor for a project i am working on and i have a structure weighing approx 2kg which is to be run at 1000 RPM. So my question is what should be specifications of the motor i am to buy ,more specificly the torque required to run the structure and the power rating? i was...
  2. evinda

    MHB Verifying the Minimum Principle for $u(x,y)$

    Hello! (Wave) Let $u(x,y), x^2+y^2 \leq 1$ a solution of $u_{xx}(x,y)+u_{yy}(x,y)+(1+x^2) e^{-u(x,y)}=0, x^2+y^2 \leq 1$. Show that $\min_{x^2+y^2 \leq 1} u(x,y)= \min_{x^2+y^2=1} u(x,y)$. Is the following right? $u_{xx}(x,y)+u_{yy}(x,y)=-(1+x^2) e^{-u(x,y)}$ $(1+x^2) e^{-u(x,y)}>0 \ \ \ \ \...
  3. MidgetDwarf

    Can relative maximum and minimum exist when the derivative....

    Can relative maximum and minimum points exist when a function is defined at say x=c, however the derivative does not exist or tends to infinity? Ie the graph of. F (x)= |x|, for x=c=o. If I am correct the relative minimum is at o, can it also be the abs minimum? I recalled the theorem by...
  4. C

    Trying to Find Coords of a Local Maximum and Minimum

    Homework Statement (c) Show that ##tan(x) + cot(x) \equiv 2csc(2x)## (d) Hence or otherwise, find the coordinates of the local maximum and local minimum points of the graph of ##y = tan(2x) + cot(2x), 0≤x≤\frac{π}{2}## Homework Equations Most likely a lot of different trigonometric formulas to...
  5. A

    Minimum uncertainty of the momentum of a small particle

    Homework Statement Find the minimum uncertainty of the momentum of a small particle with mass m=1g, which is confined within a region of width a=1cm. Homework Equations Delta(p)*Delta(x)>=hbar/2 The Attempt at a Solution Delta(p)*Delta(x)=hbar/2 Delta(p)*10^(-2)=hbar/2 Delta(p)=10^2*hbar/2...
  6. anemone

    MHB Find Minimum of y: $y=2a+\sqrt{4a^2-8a+3}$

    Find the minimum of the function $y=2a+\sqrt{4a^2-8a+3}$.
  7. T

    Minimum velocity for particle to lose contact

    Homework Statement Homework EquationsThe Attempt at a Solution Honestly I have very little idea about this problem . The contact force has to vanish for particle B to loose contact .Not sure how to relate the velocity of upper particle with the normal contact force of the lower particle ...
  8. D

    Minimum Acceleration for Safe Rope Sliding: 40kg Person on 350N Tension Rope

    Homework Statement What is the minimum acceleration with which a 40.0kg person slides down a rope that can withstand a maximum tension of 350N and have the rope not break? Homework Equations Fnet=ma The Attempt at a Solution see upload
  9. C

    Circular Aperture Diffraction, Angle of First Minimum

    Homework Statement A helium-neon laser ( ##\lambda =633nm##), is built with a glass tube of inside diameter 1.0mm. One mirror is partially transmitting to allow laser light out. From an optical perspective, the laser beam is a light wave that diffracts through a 1.0mm diameter circular...
  10. T

    Minimum force required to form sphere

    Homework Statement Homework EquationsThe Attempt at a Solution Doing a vertical force balance 2Fcosθ=mg ,where m is the mass of water . Not sure how to proceed . What role does the pin hole at the top play ? I would be grateful if somebody could help me with the problem.
  11. anemone

    MHB What is the minimum value of $f(x)$ with positive real numbers $p,q,r$?

    For positive real numbers $p,\,q,\,r$, determine the minimum of the function $f(x)=\sqrt{p^2+x^2}+\sqrt{(q-x)^2+r^2}$.
  12. J

    Find minimum number of generators for Z/nZ

    Is there a method to find the minimum number of generators ##\big<a_1,a_2,\cdots,a_k\big>## needed to generate ##\mathbb{Z}_n^*## such that ##\mathop{\cap}\limits_{n=1}^k \big<a_n\big>=\{1\}## other than by just looking for them and checking orders and products of group elements? For example...
  13. Justice Hunter

    What is the minimum speed limit in the universe?

    I have a question :x Technically, particles, like photons, which don't interact with the higgs field travel at the speed of light. wouldn't the absolute minimum speed limit be something that interacts infinitely with the higgs field?
  14. D

    Calculating minimum N so that error<10^-7 on matlab

  15. jerromyjon

    Minimum energy to accelerate a mass?

    I have a similar logical "gap" in my understanding that I still haven't resolved... which seems to be right on point with this thread if I call it "minimum amount of energy required to accelerate a given mass". From what I know about SR, accelerating mass takes energy which increases...
  16. Nick666

    The Minimum Speed in the Universe: An Exploration of the Cosmic Limit

    What is the minimum speed in the universe ?
  17. Robsta

    Show E must exceed minimum value of V(x) to be normalizable

    Homework Statement Show that E must exceed the minimum value for V(x) for every normalizable solution to the TISE, Hψ = Eψ Hint: rewrite the TISE as d2ψ/dx2 = (2m/ħ)(V(x) - E) Homework Equations I understand how to get the TISE Hψ = Eψ into the form of the hint by using the hamiltonian and...
  18. L

    MHB Very dificult: The minimum perimeter and maximum height of a triangle under constraints

    Obtain -The maximum height corresponding to the side b of any triangle (abc) once known the value of its perimeter and height corresponding to the a side a. -The minimum perimeter of any triangle (abc) once known the heights corresponding to the a and b sides. Aux: Geogebra construction...
  19. evinda

    MHB Finding a Local Minimum of $G$: Algorithm & Analysis

    Hello! (Wave) A $m \times m$ grid is a graph of which the vertices are ordered pairs of natural numbers $(n_1, n_2)$ with $1 \leq n_1 \leq m$ and $1 \leq n_2 \leq m$. Two vertices $(k_1,k_2)$ and $(k_3,k_4)$ are neighbors iff $|k_1-k_2|+|k_3-k_4|=1$. We suppose that a unique number $y_w$ is...
  20. S

    Need help finding the minimum surface area of a cylinder?

    Member warned about posting without showing an effort 1. Homework Statement 'Using the graphing function on your Graphics Calculator, or otherwise, determine the radius for a minimum Surface Area.' I HAVE NO IDEA WHAT TO DO? Homework EquationsThe Attempt at a Solution
  21. J

    Calculate minimum beam energy

    Homework Statement Calculate the minimum beam energy in a proton-proton collider to initiate the p+p→p+p+n0 reaction. The rest energy of the no is 547.3MeV.Homework Equations For a head on collision between particles a and b, from conservation of total energy: Ex = Ep + Ep' = 2Ep so that...
  22. Z

    Double Slit Diffraction:Finding Greatest Angle for Minimum

    I'm at wits end. I hate WileyPlus. Part A, B and C are correct. I cannot figure out what Part D is -- all of the answers I am getting are wrong. Homework Statement Homework EquationsThe Attempt at a Solution
  23. C

    Critical points and of polynomial functions

    Homework Statement A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the...
  24. J

    Minimum value of the electric field

    Homework Statement The potential difference Δϕ between the plates of a spherical capacitor is kept constant. Show that the electric field at the surface of the inner sphere will be a minimum at a = b/2, where a is the radius of the inner sphere and b is the radius of the outer sphere. Homework...
  25. anemone

    MHB Minimum of $a$ for $a^2+2b^2+c^2+ab-bc-ac=1$

    Let $a,\,b,\,c$ be real numbers such that $a^2+2b^2+c^2+ab-bc-ac=1$. Find the minimum possible value of $a$.
  26. CKaiL

    Find the minimum amount of energy required to eject electron

    Homework Statement Find the minimum amount of energy required to eject electron from the metal surface inside the h/e apparatus's photodiode. Homework Equations Photoelectric Effect Equation E = hf = hc/ wavelength = eV0 The Attempt at a Solution Does the metal surface inside the h/e...
  27. anemone

    MHB Minimum of the Sum of Logarithms

    Find the minimum of $\large \log_{a_1}\left(a_2-\dfrac{1}{4}\right)+\log_{a_2}\left(a_3-\dfrac{1}{4}\right)+\cdots+\log_{a_n}\left(a_1-\dfrac{1}{4}\right)$ where $a_1,\,a_2,\cdots,a_n$ are real numbers in the interval $\left(\dfrac{1}{4},\,1\right)$.
  28. D

    Minimum of two random variable

    Hello, want to know if it's correct 1. Homework Statement X and Y two random variables iid of common density f and f(x)=x*exp(-x²/2) if x≥0 and f(x)=0 if x≤0 and Z=min(X,Y) Find -The density of Z -The density of Z² - E[Z²] Homework EquationsThe Attempt at a Solution 1.[/B] FZ(u) = P(min(X,Y...
  29. S

    Find maximum wavelength using minimum wavelength and frequency difference

    Homework Statement A radar receiver can be tuned to detect signals are different wavelengths. If the frequency difference between the minimum and maximum wavelength is 10 GHz and the minimum detected wavelength is 4910µm. What would be the maximum wavelength? Homework Equations λƒ=c The...
  30. kostoglotov

    Discontinuity of a constraint in Lagrange Method

    Homework Statement My question is quite specific, but I will include the entire question as laid out in the text Consider the problem of minimizing the function f(x,y) = x on the curve y^2 + x^4 -x^3 = 0 (a piriform). (a) Try using Lagrange Multipliers to solve the problem (b) Show that the...
  31. L

    Semiconductors, minimum conductivity

    Homework Statement Show that the minimum conductivity of a semiconductor occurs when n0 = ni√(μh/μe) Derive an equation for the minimum conductivity Calculate the (i) minimum and (ii) intrinsic conductivity for silicon. State the assumptions you make and the origin of any material properties...
  32. julianwitkowski

    Calculating Minimum Values for Artificial Gravity in Centripetal Force Wheels

    Hey thanks for reading:) I'm trying to learn calc and I thought it would help if I could figure out how to do this... For school I have a physics paper on artificial gravity and how it could work... I didn't post this in HW help because It's more of a written thing and I just got sidetracked...
  33. L

    Minimum Speed for Motorcyclist to Successfully Cross a Ravine

    Homework Statement A motorcyclist must cross a ravine. The far side of the ravine is 2.0 meters higher than the launch point, a ramp that makes an angle of 40◦ with the horizontal. If the ravine is 10.0 meters wide, what minimum speed v must the biker have when leaving the ramp to successfully...
  34. anemone

    MHB Find the minimum of (4xyz)/(3)+x²+y²+z²

    Let $x,\,y,\,z$ be the lengths of the sides of a triangle such that $x+y+z=3$. Find the minimum of $\dfrac{4xyz}{3}+x^2+y^2+z^2$.
  35. Yam

    Frictional forces and the angle for minimum frictional force

    Homework Statement A person is dragging a packing crate of mass 100 kg across a rough floor where the coefficient of kinetic friction is 0.400. He exerts a force F just sufficient to keep the crate moving at a constant velocity. At what angle above the horizontal should his pulling force F be...
  36. R

    Maximum and minimum values of a variable capacitor

    Homework Statement A parallel circuit like that in the figure forms the tuning circuit for an AM radio. If the inductor has a value of 8.6×10-6 H, what must be the maximum and minimum values of the variable capacitor if the radio receives frequencies from 590k Hz to 1670k Hz? a) Maximum C? b)...
  37. P

    Minimum Force such that box m does not slide down

    Homework Statement Homework Equations F = ma; f = µN The Attempt at a Solution Let F' be the action-reaction force between box m and box M f be the friction on small box m For small box m: F' = ma f - mg = 0 µF' - mg = 0 since f = µF' µma - mg = 0 since F' = ma a = g...
  38. Albert1

    MHB How to Find the Minimum of f(x) in an Absolute Value Trigonometric Function?

    let $f(x)=\left |sin\, x+cos\, x+tan\, x+cot\, x+sec\, x +csc\, x \right |$ please find minimum $f(x)$
  39. G

    Minimum Orbital Radius Around Black Holes

    Hi there, I was reading one of my textbooks and I had a thought. For a black hole, there is minimum orbiting radius of ##R_{min}=3R_s## where ##R_s## is the Schwarzschild Radius. This minimum orbit is created by the fact that in order to obtain an orbit of that radius around a black hole, you...
  40. N

    Solving a Quadratic Equation: Finding the Minimum

    Homework Statement The Attempt at a Solution Looks like the graph would be a parabola? And since it's a>0 it would be upward and therefore a minimum. Not sure what that would be. Unsure how to solve the rest[/B]
  41. siddharth23

    Minimum engagement length of a bolt

    Hey guys. Could anyone give me an what the criteria for minimum engagement length of a bolt is? The bolt I'm using is M10x1. Thanks!
  42. J

    How to charge a low amperage battery?

    I have a motorcycle battery which is 6Ah, with a maximum charging current of 0.6A/h. Is there a method of charging it whilst using a 2.7Adc (4A RMS) charger? I was thinking using resistors in parallel or series between the charger and the battery, but I am not sure exactly how.
  43. F

    Minimum momentum of electron in a hydrogen atom

    Homework Statement The energy of an electron in a hydrogen atom is: E = p^2/2m_e - \alpha e^2/r; where p is the momentum, m_e is the electron charge magnitude, and \alpha the coulomb constant. Use the uncertainty principle to estimate the minimum momentum in terms of m_e, a, e, \hbar...
  44. D

    Minimum necessary speed for card to be able destroy glass

    Homework Statement Hello, I am international student, so sorry for possible mistakes in English. For our science night I naively choosed project, where I will try to calculate minimal necessary speed for thrown card to destroy glass (probably wine glass). There are two videos for illustration...
  45. E

    Why Velocity Same at Min Separation? | Conservation of Momentum

    Homework Statement Why is the velocity the same at minimum separation? For example, two positively charged pucks are traveling towards each other. Find the minimum separation. They oppose each other so they should slow down and reach minimum separation when their velocities are 0. It does not...
  46. L

    Effective potential energy minimum

    Effective potential energy is defined by U^*(\rho)=\frac{L^2}{2m\rho^2}+U(\rho) in many problems I found that particle will have stable circular orbit if U^*(\rho) has minimum. 1. Why is that a case? Why circle? Why not ellipse for example? 2. Is this condition equivalent with...
  47. anemone

    MHB Determine the minimum value of an expression

    If $x,\,y,\,z>0$ and $x+y+z=1$, find the minimum of $\left(x+\dfrac{1}{x}\right)^{10}+\left(y+\dfrac{1}{y}\right)^{10}+\left(z+\dfrac{1}{z}\right)^{10}$.
  48. ubergewehr273

    What is the Most Effective Method for Minimum Stopping Distance in a Car?

    Homework Statement In order to stop a car in shortest distance on a horizontal road, one should, (a) Apply brakes very hard so that the wheels stop rotating (b) Apply the brakes hard enough to just prevent slipping (c) Pump the brakes (press and release) (d) Shut the engine off and not apply...
  49. M

    Minimum time window needed to capture frequencies

    I'm pretty sure there have to be some theorems out there, but I am probably not putting in the right search terms to be able to find them. Here's the problem: I have a signal uniquely composed of a finite summation of standing wave sinusoids (well there's some DC and other background, but let's...
  50. S

    Minimum angle the ladder can make with the floor without slipping

    Homework Statement [/B] A 3.0-m-long ladder, as shown in the following figure, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is 0.32. What is the minimum angle the ladder can make with the floor without slipping? Homework Equations...
Back
Top