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. J

    Find the supremum, infinum, maximum and minimum

    Homework Statement find the supremum, infinum, maximum and minimum Homework Equations (n-2sqrt(n)) n is element of natural numbers The Attempt at a Solution no idea on how to do this help please
  2. K

    Minimum velocity to change radius of orbit

    Homework Statement A rocket is in an elliptical orbit around the earth; the radius varies between 1.12 Rearth and 1.24 Rearth. To put the rocket into an escape orbit, its engine is fired briefly in the direction tangent to the orbit when the rocket is at perigee, changing the rocket's velocity...
  3. V

    Maximizing Base Angles for a Fixed Area and Base Length

    Homework Statement The base b and the area k of a triangle are fixed. Determine the base angles if the angle at the vertex opposite b is to be a maximum. Homework Equations The Attempt at a Solution The answer the book gives is arctan(4k/b2) I looked at the solution to the...
  4. F

    Minimum Separation of Two Frictionless Pucks

    two frictionless pucks are placed on a level surface, as shown, at an initial distance of 20.0 m. the mass of puck 1 is 0.80 kg and it has a charge of +3.0 X 10^-4 C., while puck 2 is 0.40 kg with a charge of +3.0 X 10^-4 C. the initial velocity of puck 1 is 12.0 m/s (E) and the initial velocity...
  5. R

    Absolute Maximum and Minimum Values over a given interval

    Homework Statement Find the absolute maximum and absolute minimum values of the function (x^3)+(12x^2)-27x+9 on each of the following interval [-10,0] The Attempt at a Solution I got this and its saying its wrong. y'= (3x^2)+(24x)-27 factor: 3(x^2+8x-9) so x=-9 and x=-1 y''=6x+24...
  6. R

    Maximum And Minimum of a function

    Function:x+x2/3 I got 0 and -(8/27) as the criticical numbers. Am I right? I also want to know if this function as a max and a min. Note:2/3 is an exponent.
  7. E

    Finding Min Velocity for Projectile from A to B

    Homework Statement What I would like to do is find the minimum velocity needed to launch a projectile from point A to B when B is not at the same height as A. I keep finding myself stuck with two unknowns (initial speed and release angle) and I am not sure how to resolve them. If the problem...
  8. M

    Trigonometry, find the minimum of tan(a).tan(b).tan(c)

    Homework Statement 1. Given a,b,c are acute angles and a + b + c =180. find the minimum of tan(a).tan(b).tan(c) 2. Prove that if a+b+c=90, then tan(a)+tan(b)+tan(c) >= 31/2 Homework Equations The Attempt at a Solution I don't even have any ideas how should i start to find/prove them... any...
  9. C

    Minimum velocity for a sonic boom?

    On our grade 11 physics exam, there was a question asking for the minimum velocity of a jet in order for it to produce a sonic boom (the temperature of the air was given, so the speed of sound in the situation was known). According to our textbook, a sonic boom is produced only after the jet's...
  10. L

    Equilibrium: Minimum Cable Length

    Homework Statement The above figure shows a uniform iron beam of mass 253 kg and length L = 3 m. The cable holding the beam in place can take a tension of 1300 N before it breaks. Use the Young's Modulus for steel to be 200e9 N/m2. (You may ignore the small mass of the cable in this...
  11. G

    Hover bike - what is the minimum propeller rpm required?

    Hello, I read this article about someone who made a hover bike prototype from a motorbike and two propellers. Best part is, its classified as Ultralight and doesn't require a license to fly! I can totally see myself building something like that in my back garden! How can I calculate the...
  12. G

    Minimum cost of power line

    Hey guys I know this questions has been posted before but I still can't seem to figure it out. Thanks a lot for your help (: Homework Statement A power line is to be constructed from a town to a mine that is located near a straight road going to the town. The point on the road closest to...
  13. J

    Find minimum energy of a photon to produce 25keV electron in compton scatter?

    Homework Statement What is the minimum energy that photons must have to produce 25 keV electrons in a Compton scattering? (answer in keV) Homework Equations Not too sure here, We know Eincident= Efinal + K.E.electron We know that the work function (binding energy) is also taken into...
  14. C

    Question: Minimum uncertainty in ground state electrons

    Hey all, I'm a student curently studying in a Singaporean Junior College (American 12th grade equivalent). I was curious and just thought of the following: Suppose a system with a electron and a proton nucleus, a hydrogen atom. From electrostatic force and circular motion equations- F =...
  15. R

    Minimum radius to avoid voltage breakdown?

    I am trying to figure out the minimum radius needed to avoid voltage breakdown. I found this from a Physics website: "The electric field near a conductor is inversely proportional to the radius of curvature of the surface." So if I know the voltage and the distance between the 2 metal...
  16. J

    Maximum and Minimum in Trigonometry

    Homework Statement Find Maximum and Minimum value of Homework Equations \mathbf{f(x) = sin^{2n+1}\; x - cos^{2n+1}\; x} and \mathbf{n\in\mathbb{N}} The Attempt at a Solution put \mathbf{n=1, f(x) = sin^3\; x -cos^3\; x}\\\\ get \mathbf{-1\leq f(x)\leq 1}
  17. B

    Finding the Minimum of a Function: A Guide for Calculus Students

    Homework Statement Check for minimum: When you've got your x value and sub it back into the f'(x) equation, should you get zero if it's a minimum? Homework Equations The Attempt at a Solution
  18. 7

    Mathematica Finding Minimum Coefficient of (1+x)^n for k<=20

    Hello every one, I wish i can some help in my mathematica programing. I hane k <= 20, k is + integer. n = 1/12(3(5 - 2Sqrt[6])^k) what I need to do is for every value of n (comes from K ofcourse), Print the minmum coefficient for (1 + x)^n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%...
  19. J

    Minimum area foot land glider

    I am looking into how small (minimize surface area) a glider could be to be foot land. I would like to look into possibly designing and or building a backpack deployable glider, but first would just like to design a non-deployable glider with minimal surface area. To do this, currently my...
  20. Femme_physics

    Finding the minimum length of a median in a triangle

    Homework Statement In a right triangle ABC, AD is a Median to BC (see drawing) Given: AB+BC = 4 A) Mark BD with an x, and express the lengths of the adjacent and opposite through x. B) Find the x for which the length AD is minimal http://img641.imageshack.us/img641/6798/abctriangle.jpg The...
  21. B

    Speed of sound minimum distance problem

    Hi all, just a question I am having trouble with 1.) You're watching a carpenter pound a nail. He hits the nail twice a second, but you hear the sound of the strike when his hammer is fully raised. What is the minimum distance from you to the carpenter? assume the air temperature is 20...
  22. S

    Finding minimum distance between two points on two different parabolas

    Hi everyone. I've got a homework problem with Differential Calculus that drives me crazy for a couple of days so far and would like to ask you guys for a few suggestions Homework Statement Given two parabolas, (C1): y = x2 and (C2): y = - (4-x)2. Find two points, each point on one parabola...
  23. A

    Find the minimum distance from a curve to a point not on it

    Homework Statement You are given point (1,2). f(x) = 0.199445983x^2 - 3.789473684x + 19 f'(x) = 0.398891966x - 3.789473684 You also know (0,19), (9.5,1) <-- this is the vertex. This graph has the domain 0≤ x ≤ 9.5 Find the point on the curve that when joined to a line segment with...
  24. N

    What orientation will give the minimum mutual inductance?

    Homework Statement Suppose a fixed distance separates the centers of two circular loops. What relative orientation of the loops will give the maximum mutual inductance? What orientation will give the minimum mutual inductance? Homework Equations none The Attempt at a Solution...
  25. K

    Minimum Work Needed to Bring 2 charges from distance infinity

    Two charged particles are brought together from a great distance (r=infinity) to a distance of 2 m. The particle has a positive charge of 3.0 x 10^-5 C and the second has a negative charge of 1.35 x 10^-5 C. What minimum work was accomplished in this process? I can't for the life me figure...
  26. Z

    Temperature of minimum density

    What is the temperature at which water possesses minimum density?
  27. B

    Find the minimum value taken by the integral expression

    Homework Statement find the minimum value taken by the integral expression integral from 0 to pi/2 [(y')^2]-(y^2)+2xy dx y(0)=0 y(pi/2)=pi/2 The Attempt at a Solution using euler lagrange d/dx(2y')-(-2y+2x)=0 2y''+2y-2x=0 from here I'm stuck
  28. J

    MATLAB Optimizing Minimum Value Search in MATLAB: A Guide for Scientists

    I got a quick question, let's say I have an array of numbers for example [ 9 3 2 7 1 8 ] and I want to find the minimum value ie: for this example it would be 1. How could I implement an optimization search like this??
  29. K

    Minimum time through a cyclotron

    Homework Statement v initial is also equal to 1000 Homework Equations ? The Attempt at a Solution the only time the particle actually speeds up is during the crossing of the dees. so integral of of Vdx = the maximum energy of the system, but i don't know how to make this into an...
  30. S

    What is the Minimum Detectable Power of a Light Flash on a Silicon Solar Cell?

    Homework Statement A circuit employs a silicon solar cell to detect flashes of light lasting .25 seconds. The smallest current the circuit can detect reliably is .42 \muA. Assuming that all photons reaching the solar cell give their energy to a charge carrier, what is the minimum power...
  31. W

    Find Max/Min of |f(z)| at z=2: Maximum Modulus Theorem & Minimum Modulus Theorem

    Homework Statement Suppose that f(z)=(z-1)(z-4)^2. Find the lines through z=2 on which |f(z)| has a relative maximu, and those on which |f(z)| has a relative minimum, at z=2. Homework Equations Maximum Modulus Theorem And Minimum Modulus Theorem The Attempt at a Solution So I know...
  32. D

    Finding minimum speed given position vector.

    Homework Statement r(t) = < t^2, 6t, t^2 − 24t > Using this position vector, find the minimum speed of the particle. Homework Equations v(t) = r'(t) The Attempt at a Solution I've found similar topics with similar problems but I'm having a hard time figure mine out. I know...
  33. K

    Coefficient of static friction and minimum horizontal force

    Homework Statement A 5.0 kg mass is held against a wall with which it has a coefficient of static friction of 0.85. What is the minimum horizontal force required to hold the mass against the wall ? The teacher said answer is 58 N .. i can't get to that answer.. Homework Equations...
  34. C

    Rotation - Minimum force so that the block topple

    Homework Statement The Attempt at a Solution I have worked my brain out but can't work out the problem the normal force will act through th center of mass right? if not please some one tell me why !
  35. T

    Finding the minimum angle of deviation for an equilateral prism

    Homework Statement Use the formula: n=(sin(A+D)/2)/(sin(A)/2) to find D for an equilateral prism of index of refraction n=1.50 (where A=apex angle for a prism=60 degrees in this case, and D=minimum angle of deviation) Homework Equations Snell's Law: (n1)sin(i)=(n2)sin(r)...
  36. V

    What is the maximum period of revolution for a cone filled with water?

    Homework Statement A SMALL CONE FILLED WITH WATER OF RADIUS 4M AND THE WATER DOESNT FALL DOWN. WHAT MUST BE THE MAXIMUM PERIOD OF REVOLUTION Homework Equations The Attempt at a Solution I AM CONFUSED! HOW TO APPROACH THE PROBLEM? SHOULD FIRST CALCULATE THE MINIMUM VELOCITY...
  37. B

    Find the Minimum Volume of cone, answer does not make sense

    Homework Statement Find the Dimensions of the right circular cone of minimum volume which can be circumscribed about a sphere of radius 8 inches.Homework Equations N/A The Attempt at a Solution So this is my try, i did the question to find the minimum volume of the cone, For Larger Size...
  38. E

    Solving a Balloon Mass Calculation: Find Minimum Gas Mass Needed

    Homework Statement A balloon contains gas of density pg and is to lift a mass M, including the balloon but not the gas. Show that the minimum mass of gas required is m= Mp / (pa - pg). Where pa is Atmospheric density Homework Equations I assume fluid motion so P= Po + p*h*g Also...
  39. B

    Find the minimum premeter for a rectangle?

    Homework Statement A rectangular field to contain a given area is fenced off, along a stright river. if no fencing is required along the river, show that the lest amount of fencing will be required when the length of the field is twice its width. Homework Equations N/A The Attempt...
  40. B

    Finding the x that makes y minimum in MATLAB.

    Hello. As stated in the title, I have a function y that is in terms of x. How do I go about determing the 'x' that makes 'y' a minimum? Thanks!
  41. E

    Minimum Momentum and Energy from Schrödinger Equation

    Homework Statement I have a 2D square well with sides a. What is the minimum momentum and energy for this well? Homework Equations Minimum momentum given by Δp≈ℏΔx and Δp≈ℏΔy Minimum energy given by ΔE≈ℏΔt This might be right. It says to consider the Schrödinger Equation in 2D. I've never...
  42. B

    Prove that the gradient is zero at a local minimum.

    Homework Statement Suppose F: Rn --> R has first order partial derivatives and that x in Rn is a local minimizer of F, that is, there exists an r>0 such that f(x+h) \geq f(x) if dist(x, x+h) < r. Prove that \nabla f(x)=0. Homework Equations We want to show that fxi(x) =0 for i = 1,...,n So...
  43. D

    FFT minimum sampling duration

    I am wondering what the minimum sampling duration must be in order to accurately resolve a 60Hz wave. I know the FFT works with periodic waves about the number N but I am not sure how this relates to say a sampling duration that is 1 half the sampling period of a 60 Hz wave. I'm worried that the...
  44. S

    Langrange Multipliers and Minimum Distance

    Homework Statement Use the method of Lagrange multipliers to find the points on the ellipse (x^2)+(2y^2)=2 that are closest to and farthest from the line x-y=2. Homework Equations I know that to do the lagrange I have to take the gradient of a function f and set it equal to some unknown...
  45. N

    Finding the minimum area of a triangle

    So i have the points A=(1,0,1) and B=(1,-1,0) and the third corner lies on the line L:(x,y,z) = (t,t,t) and i need to find a triangle with the minimum area possible. My initial approach was to calculate the vector BA and BC and then apply both vectors in the area triangle formula...
  46. R

    What is the minimum work needed to cool an object using a carnot refrigerator?

    Homework Statement 433J of heat is extracted from a massive object at 0\circC while rejecting heat to a hot reservoir at 19\circC. What minimum amount of work will accomplish this? Answer in units of J. Homework Equations COP= Qc/(Qh-Qc) = Tc/(Th-Tc) The Attempt at a Solution...
  47. A

    Minimum potential energy of a simple harmonic oscillator

    Homework Statement A simple harmonic oscillator of force constant 2*106 N/m and amplitude .01 m has total maechanical energy 160 J... Homework Equations The Attempt at a Solution Now this is not the question but what is the minimum potential energy...1/2kx^2 comes out to be...
  48. G

    Shape of Minimum Moment of Inertia for Fixed Volume: How to Prove?

    Homework Statement Find the shape of the solid of revolution with the minimum moment of inertia about it's axis, given the volume is fixed. Homework Equations I=\int\frac{1}{2}r^2dm=\int\frac{1}{2}r^2\rho~dV=\int\frac{\rho\pi}{2}r^4dz V=\int \pir^2dz I+\lambda V must be stationary. The...
  49. 5

    What is the Optimal Speed for Minimizing Coal Consumption on a Ship?

    I cannot seem the crack this problem. The solution may be trivial, but normal methods of max/min do not seem to work. Thanks for your help. Homework Statement Suppose it to be known that consumption of coal by a certain ship may be represented by the formula y = 0.3 + .001v3 Where y...
  50. F

    Minimum strength of pipe for load bearing

    Hello, I am working on a project and I came across a question that is a little beyond my skill set. I am designing a support for a roof, and I have the force into the post, the length of the post and I need to determine what pipe I need, size, schedule diamter, etc. How do I go about this...
Back
Top