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. S

    STATICS problem: Minimum Resultant Force

    Problem Statement: If the resultant force acting on the bracket is required to be a minimum, determine the magnitude of F1 and the resultant force. Set ∅=30 deg. I have found the components of each force and set them up into a resultant vector, with F1 solved symbolically into its...
  2. K

    Why should i=e for minimum deviation in a triangular prism?

    When I was solving questions on Ray optics I encountered a question wherein I had to prove that when a ray of light undergoes minimum deviation through a triangular prism then both the angles of refraction that is the first one at the incident point and the second one at the emergent point...
  3. B

    Minimum d occurs at minimum d^2

    Homework Statement Find the point on the parabola y^2 = 2x that is closest to the point (1,4) Homework Equations d = ((x-1)^2 + (y-4)^2)^1/2 x=y^2/2 The Attempt at a Solution My attempt involves subbing in x in the d equation and differentiating it. I can't get the same answer as the book...
  4. V

    Find the Minimum Length of a Vibrating String

    Homework Statement the equation of a stationary wave produced on a string whose both ends are fixed is given by y= [0.6sin(pi/10)x]cos(600pi)t what could be the smallest length of string?Homework Equations k=2pi/wavelength The Attempt at a Solution i got the wavelength to be 20 but don't know...
  5. Brian T

    Best Place to View Stars: North Pole

    From my astrophysics class, we were given that the minimum altitude of a star relative to your latitude on Earth (\phi) is given by: $$h_{min} = \delta + \phi - 90°$$ (where \delta is the declination angle of the star) Supposing you were trying to view as many stars as possible in the night sky...
  6. O

    Minimum work to separate O2 &N2 from air

    1. assume air is an iseal gas at 25C and 100kPa. the mole fractions of components are nitrogen_0.7567 oxygen=0.2035 water vapor=0.0303 carbon dioxide=0.0003 Calc. min work to seperate a-1kmol O2 at 25C 100kPa b-1kmol N2 at 25C 100kPa from atmosphere Homework Equations...
  7. Albert1

    MHB Find Min Value of a+b+c=1: $\sqrt {a^2+b^2}+\sqrt {b^2+c^2}+\sqrt {c^2+a^2}$

    $a>0,b>0,c>0 ,\,\, and \,\, a+b+c=1$ find $min(\sqrt {a^2+b^2}+\sqrt {b^2+c^2}+\sqrt {c^2+a^2} )$
  8. Albert1

    MHB Minimum Perimeter of a Trapezoid: Find R & P

    a trapezoid $ABCD,$ with $\overline {AD}// \overline {BC}, \overline {AB}=\overline {CD}$, and diagonal $\overline {AC}=15=\overline {BD}$ if R is its maximum area ,please find : (1)R (2)find its minimum perimeter P
  9. B3NR4Y

    Proof that an even degree polynomial has a minimum

    Homework Statement Let $$p(x) = a_{2n} x^{2n} + ... + a_{1} x + a_{0} $$ be any polynomial of even degree. If $$ a_{2n} > 0 $$ then p has a minimum value on R. Homework Equations We say f has a minimum value "m" on D, provided there exists an $$x_m \in D$$ such that $$ f(x) \geq f(x_m) = m $$...
  10. Q

    Minimum work to transport electron?

    Homework Statement A charge Q = -820 nC is uniformly distributed on a ring of 2.4 m radius. A point charge q = +530 nC is fixed at the center of the ring. Points A and B are located on the axis of the ring, as shown in the figure. What is the minimum work that an external force must do to...
  11. P

    Minimum magnitude resolvable by a spectrograph

    I am stuck on a problem regarding the minimum magnitude of an astronomical object that can be handled by a given telescope+spectroscope combination. The telescope aperture (D_T) is 2 m wide, the exit beam (D1) is 3 cm wide. The slit width (s) is 10 micrometers. The wavelength of interest is 450...
  12. TheMathNoob

    Algorithm to find the minimum spanning tree on a planar graph

    Homework Statement I have the following algorithms: Prime MST with array, Prime MST with priorityq and Kruskal with priorityq. I have to choose the best algorithm to find the minimum spanning tree on a planar graph. A planar graph has the property in which the number of edges is in O(number of...
  13. W

    Minimum Angle to Prevent Slip

    Homework Statement 9.12) The mass of the box on the left is 30kg and the mass of the box on the right is 40kg. The coefficient of static friction between each box and the inclined plane is μs = 0.2. Determine the minimum angle α for which the boxes will remain stationary. Homework Equations...
  14. M

    How can I find the minimum index of refraction?

    Homework Statement We look at the center of one face of a solid cube of glass on a line of sight making 55° with the normal to the cube face. What is the minimum refractive index of glass for which you will see through the opposite face of the cube? (Hint: see through will be possible if the...
  15. Ali Durrani

    Minimum Ignition Energy of HD-5 LPG

    The minimum ignition energy for propane is 0.25mJ and for propene is 0.28mJ i have a mixture of propane and propene the percent by volume of propane is 95% and of propene is 5%, is there anyway other than experimentation to find the exact value of Minimum ignition energy of this Mixture or if...
  16. patrickbotros

    A Find the minimum without Calculus or Graphing

    ƒ(ß)=.5sec(ß) + √[1+(sec2(ß)/4)+tan(ß)/√(2)] Without graphing it or using calculus find the minimum. I already know the answer but want to know how to do it. It s at π/12 and is something like 1.5. First off this is NOT a homework problem. I already know the answer is something like 1.5 at π/12...
  17. anemone

    MHB Find the maximum and minimum of an expression

    Find the maximum and minimum of $x^3+y^3+xy(x^2+y^2)$ given $x,\,y$ are two non-negative real numbers that satisfy $x+y+xy=3$.
  18. tasleem moossun

    Evaluating the location r_m of the minimum of potential

    Homework Statement A gas of neutral atoms is often modeled as an ideal gas, where the gas atoms are considered to be "elastic billiard balls" that only interact by bouncing off each other in a manner that conservers the total kinetic energy. Homework Equations V(r) = 4(Epsilon)[(delta/r)^12 -...
  19. anemone

    MHB Find Min of x+y+z for Real x,y,z ≤ 3/2

    Find the minimum of $x+y+z+\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}$ for all positive real $x,\,y$ and $z$ that satisfies the condition $x+y+z\le \dfrac{3}{2}$.
  20. I

    Finding maximum and minimum values of 3 dimensional function

    Homework Statement Find the maximum and minimum values of f(x,y,z) = x^2 - 2x + y^2 - 4y + z^2 - 4z in the region x^2 + y^2 + z^2 <= 36. Homework Equations Lagrange multipliers The gradient of f = lambda * gradient of g <Fx, Fy, Fz> = lambda<Gx, Gy, Gz> Test for local extrema If Determinant...
  21. Calpalned

    Calculate the minimum mass of an X-ray binary

    Homework Statement Homework Equations n/a The Attempt at a Solution I don't know how to start part A and part B depends on part A. How does one calculate a lower mass limit? Is it the lower limit of an integral? I also don't understand how there can be two periods (one at 0.714 seconds...
  22. C

    Minimum distance to avoid collision using relative motion

    Homework Statement 2 cars with velocities 10 m/s and 20 m/s are traveling in opposite directions, having uniform retardation of 2m/s^2 and 1m/s^2 respectively. Find minimum separation between them such that they don't collide. Homework Equations $$v^2=u^2 + 2as$$ $$V_{12}=v_{1}-v_{2}$$ The...
  23. P

    Other Next Leonard's Susskind's theoretical minimum

    Do you guys know wether Leonard Susskind is going to release the next theoretical minimum book which is expected to be focused on special relativity and electrodynamics?
  24. C

    MHB How can I show b^2 > 24c in a cubic with one maximum and one minimum?

    Hello, been thinking on this one for a little while, and can't seem to figure it out. Problem statement is: The cubic curve y = 8x^3 + bx^2 + cx + d has two distinct points P and Q, where the gradient is zero. Show that b^2 > 24c. It seems simple enough, but I can't logic it out. This...
  25. W

    How Do I Start Solving Part C of the Physics Problem?

    Homework Statement is part C lack of information ? can someone give me a hint how to start ? for part a , -(1.35-0.95)(200cos60)-2(200sin60)=-386N(anticlockwise) for part b , -386.4/2 = 193N (in upward direction) Homework EquationsThe Attempt at a Solution
  26. A

    Minimum and Maximum Observed Frequencies?

    Homework Statement Let's say you want to find the minimum and maximum observed frequencies of light observed on Earth from the sun, when considering the Earth's motion around the sun. How would you go about doing this if you assume Earth takes time X to complete a revolution around radius R...
  27. Nouo

    Calculating minimum velocity around a loop

    So I had this question for physics about something going around a loop. The question asked what the minimum velocity to counteract gravity is if something is going around a loop. hint: One force will be zero. I'm not sure how to figure this out, can someone help?
  28. A

    What is the minimum safe following distance at highway speeds?

    Homework Statement you are driving at highway speeds of 32m/s. The car in front of you suddenly brakes with an acceleration of 1.86 m/s^2. If your car can brake with an acceleration of 1.76m/s^2. What is the minimum following distance that you need before the braking in order to avoid a...
  29. J

    Minimum output e.m.f. of the power source

    Homework Statement Homework Equations V=IR For resistors connected in parallel, 1/R =1/R1 + 1/R2 +... The Attempt at a Solution I first calculated the resistance of the the cable which consists of two wires, 1/R= 1/0.0050 + 1/0.0050 R = 2.5x10^-3 x 800 = 2 Ω Then I calculate the voltage...
  30. M

    MHB Is the Variational Principle an introductory example for Calculus of Variations?

    Hey! :o We want to get from the poit $A$ to the point $B$ avoiding the lake. How should we move to minimize the distance? There are the following possibilities: 1. (We go along the red lines) In this case the distance is the following: Let $y_1$ be the first red line and $y_2$ the...
  31. P

    Find minimum value for coefficient of static friction

    Homework Statement Homework Equations Newton's 2nd Law The Attempt at a Solution I was wondering if anyone could verify that I have all the correct vectors or if I am missing any. I just wanted to check before I started trying to derive the equations. [/B]
  32. L

    How to get (v1+v2)/n for odd n with minimum resistors

    For example, the simplest case n=3 I am even stuck on much less the general problem we can start with that and evidently it may be able to be soled with only two or three resistors if you are clever Also we are not allowed to use op amps only resistors and we are given two voltage sources...
  33. mooncrater

    Minimum and maximum number of bits required....

    Homework Statement Here is this line in my book which says, "Although the minimum number of bits required to code ##2^n## distinct quantities in n, there is no maximum number of bits that may be used for a binary code. For example, the ten decimal digits can be coded with ten bits, and each...
  34. chuttney1

    Trying to Calculate The Minimum Force For Skateboard.

    I'm building an electric skateboard. I am trying to calculate the bare minimum force and torque to move a 180 lb object on 4 polyurethane wheels from standstill for motor selection. I got lost in finding the coefficient of friction for polyurethane wheels and whether I am using the equation for...
  35. anemone

    MHB Find Min Value of $a^2+b^2+c^2+d^2+(a+b+c+d)^2$ w/ Given Constraint

    Find the minimum of $a^2+b^2+c^2+d^2+(a+b+c+d)^2$ given $a+2b+3d+4d=1$ for all real $a,\,b,\,c$ and $d$.
  36. Hyo X

    Wirebonding - minimum contact area

    I am designing a mask for a chip which might be used for wirebonding. What is the minimum contact area for wirebonding using a microscope and microprobe? Currently my contact pads are 220 x 220 um. Is that big enough?
  37. D

    Minimum initial velocity for throwing a ball off a hemispherical rock

    Homework Statement At which minimum velocity should you throw the ball horizontally if you are standing on a hemispherical rock of radius R so that it at no point touches the rock and lands at the minimum distance from the rock horizontally. Find the expression that solves for initial velocity...
  38. B

    Does Minimum of Complex Set Subset Exist?

    Homework Statement The following doesn't come from a textbook, and I am very uncertain whether it is true or false. Suppose that ##B \subseteq \mathbb{C}## is a convex set, and consider the set ##L_B := \{|b|: b \in B \}##. Homework EquationsThe Attempt at a Solution My question is, will...
  39. bland

    Minimum possible warning for deadly comet/asteroid

    What is the minimum possible warning that we on Earth can get for a very destructive space object from striking the Earth, say something about a kilometre in size (I'm guessing that would be big enough to be noticed all over the Earth). By minimum possible, I mean with the way the skies are...
  40. M

    Distance between atoms - Minimum energy

    Homework Statement The effective potential between two atoms of same mass m is: V(x)=-a\frac{1}{x}+b\frac{1}{x^2} where a,b>0 and x is the distance between them. (a) Calculate (order of magnitude) the distance between the two atoms in the molecule and its minimum possible energy. (b)...
  41. D

    Banked Curve - Minimum Turn Radius

    Homework Statement A car traveling 10 m/s is moving along a track banked at 5 degrees. The tire-road friction coefficient is .3 What is the minimum radius it can travel without slipping? v0= 10 m/s Bank Angle Θ = 5° μ=0.3 Note I am working through prep material for the exam. The solution...
  42. anemone

    MHB Find maximum and minimum of P

    Find the minimum and maximum of $P=\dfrac{y−x}{x+8y}$ for all real $x$ and $y$ that satisfy the equation $y^2(6-x^2)-xy-1=0$.
  43. B

    Derivation of Helmholtz Energy Minimum in Thermodynamics

    Hey, The helmholtz energy is supposed to have a minimum when the entropy has a maximum value. Does anyone knows the derivation for this statement?
  44. A

    Adiabatic filling of a tank - Minimum work required

    Hi guys, need an help about that problem That's an adiabatic filling of a reservoir (tank) from a line. Line is at 350 K and 8 MPa. The tank is, in the final state, at 365 K and 6 MPa. Tank is insulated (no heat exchanged during the process), and only one stream enter in the tank. How can i...
  45. C

    Determining Minimum Height to Complete a Loop

    This question was posted on Khan Academy. I attempted it after watching all of their great Work and Energy videos, but the way they arrived at the answer was way above my skill level even after reviewing the videos a number of times. Here is the question: What is the minimum height h of a slope...
  46. Math Amateur

    MHB Kunz - Vanishing Ideal and Minimum Polynomial

    I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with some aspects of Kunz' definition of the vanishing ideal of an algebraic curve and Kunz' definition of a minimal polynomial ... The relevant text from Kunz is as...
  47. A

    Calculating Minimum Possible Acceleration from given v and d

    Basically what I want to do is the opposite of finding average velocity from constant acceleration. I want to know how to find the minimum possible acceleration where the average velocity v is known and the distance d is know. The acceleration is assumed to be constant. For for example, a ship...
  48. Albert1

    MHB Find the Minimum Value of a Complex Equation

    $a>1,b>1$ find the minimum value of $\dfrac {a^2}{b-1}+\dfrac {b^2}{a-1}$
  49. E

    Electric Field and minimum velocity

    Homework Statement Electric field given by the vector E = E0/l (xi + yj) N/C is present in the xy plane. A small ring of mass M carrying charge +Q, which can slide freely on a smooth non-conducting rod, is projected along the rod from the point (0,l) such that it can reach the other end of the...
  50. J-dizzal

    Static equilibrium minimum cylinder mass problem

    Homework Statement Determine the minimum cylinder mass m1 required to cause loss of contact at A. Homework Equations ΣF=0, ΣM=0 The Attempt at a Solution Im not sure where to go with this one. the solution would be in terms of mass(m)
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