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. J

    Show rectangular box of given volume has minimum surface area when

    Homework Statement show rectangular box of given volume has minimum surface area when the box is a cube [gotta show it with partial derivatives to minimize] Homework Equations surface area = 2(wl+hl+hw) volume = whl The Attempt at a Solution so this is the one I would be minimizing...
  2. anemone

    MHB Minimum value of $a$ for Quadratic Polynomial

    The quadratic polynomial $ax^2+bx+c$ has two distinct roots $p$ and $q$, with $a,\,b,\,c$ are positive integers and with $p>0$ and $q<1$. Find the minimum possible value of $a$.
  3. B

    Taylor Series Expansion About a Local Minimum

    Hello everyone, I am currently reading chapter two, section 3 of Griffiths Quantum Mechanics textbook. Here is an excerpt that is giving me some difficulty: "Formally, if we expand V(x) in a Taylor series about the minimum: V(x) = V(x_0) + V'(x_0) (x-x_0) + \frac{1}{2} V''(x_0)(x-x_0)^2...
  4. V

    Friction - minimum value of force for movement

    Homework Statement Determine the minimum value of the force F for which the systems start moving. Assume M1 = 1kg, M2 = 2kg, Us = 0.3. Solution: F = 9; F = 6N My problem is that i don't know the forces acting on wheel. Homework Equations The Attempt at a Solution M1 : F -...
  5. L

    What Are the Maximum and Minimum Values of y When x^3 is 8+- 2(14)^1/2?

    Homework Statement Show that the maximum and minimum values of y occurs when x^3=8+- 2(14)^1/2 Homework Equations The Attempt at a Solution
  6. I

    Problem with Minimum Nyquist bandwidth formula and m-ary encoding

    I am using the book Electronic Communication Systems by Wayne Tomasi: I had a problem with a certain part of the book: The book gives the formula for the minimum nyquist bandwidth as: " The minimum theoretical bandwidth necessary to propagate a signal is called the minimum Nyquist bandwidth...
  7. Y

    MHB Local Min Problem: Solve 2a>0 & 4a2-1 > 0?

    Hello all, I have this tricky question, I think I got the idea, just wish to confirm. If the function \[z=x\cdot ln(1+y)+a(x^{2}+y^{2})\] has a local minimum at (0,0), then: (choose correct answer) 1) a<-0.5 2) a>0 3) a>0.5 4) -0.5<a<0.5 5) a>0.5 or a<-0.5 What I did, is calculate the...
  8. belliott4488

    Appropriate distribution for minimum distance

    I have what I think is probably a basic question from probability and statistics (about which I'm pretty ignorant). If I have a set of projectile trajectories that were generated by a Monte Carlo process, and I'd like to know the probability the projectile will come within distance d of some...
  9. M

    Minimum Value of Particle in Space

    Homework Statement At what time t does the speed of the particle moving in space with its position function r(t)=##<t^2, 3t, t^2 - 8t>## have its minimum value? Homework Equations Derivative, speed The Attempt at a Solution Found derivative. r'=<2t, 3, 2t-8> Found speed...
  10. S

    Minimum distance for annihilation

    How close does does a particle and anti-particle pair have to be with each other in order to achieve annihilation?
  11. anemone

    MHB Find the minimum of x+y+xy and x+y-xy

    Hi MHB, I've one problem that I think I've already solved half of it, but fact is I really don't know if I am on the right track... that problem is hurting my head so much... Problem: For all positive real $x$ and $y$, find the minimum of $x+y+xy$ and $x+y-xy$ if $(x+y+xy)(x+y-xy)=xy$...
  12. W

    How to find minimum turning points

    I have a function: f(x) = Asin2(x) + Bcos2(x) + Csin(2x) and I want to find the minimum turning point(s). To start with, I calculated: f'(x) = (A - B)sin(2x) + 2Ccos(2x) Therefore, turning points occur when f'(x)=0, or: tan(2x) = 2C / (B - A) To find the minima, I then want to...
  13. U

    Find minimum value of the expression

    Homework Statement Let n be a positive integer. Determine the smallest possible value of $$|p(1)|^2+|p(2)|^2 + ...+ |p(n+3)|^2 $$ over all a monic polynomials p with degree n. The Attempt at a Solution Let the polynomial be x^n+c_{n-1} x^{n-1} +...+ c_1x+c_0 p(1) =...
  14. Nathanael

    Minimum Velocity of an object thrown directly up to never fall down?

    EDIT: This is in the wrong section isn't it? How do I move it to the General Physics section? (My bad.) This isn't a homework problem (I'm not in a physics class) so hopefully this isn't the wrong section. My question is about finding the minimum velocity needed for an object (directly...
  15. anemone

    MHB What is the minimum value of f(x,y) under a symmetric constraint?

    Let $m,\,n,\,p,\,q$ be the positive real numbers such that $\dfrac{1}{1+m^4}+\dfrac{1}{1+n^4}+\dfrac{1}{1+p^4}+\dfrac{1}{1+q^4}=1$, find the minimum of the product of $mnpq$.
  16. thejackal

    Get the minimum taken to move from A to B given a cap on Velocity

    Homework Statement Find the minimum time taken for the object to cover the distance x I'm given the following: x = the distance i need to cover V0 = the initial velocity which is zero a = the maximum permissible acceleration Vmax = the maximum permissible velocity Homework...
  17. M

    Why do functions have maximum and minimum values in intervals?

    Hello, This might seem like a very basic problem for the most of you but I am a little bit confused as to the problem of maximum and minimum values. For, example I do understand why y = x^2 has no absolute maximum as y-->\infty as x-->\infty... However, in the closed interval, say, [0, 2]...
  18. anemone

    MHB Find Minimum Value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$

    Find the minimum value of $(x-y)^2+\left( \sqrt{2-x^2}-\dfrac{9}{y} \right)^2$ for $0<x<\sqrt{2}$ and $y>0$.
  19. U

    What is the minimum number of red balls in the bag?

    Homework Statement A bag contains n identical red balls, 2n identical black balls and 3n identical while balls. If probability of drawing n balls of same color is greater than or equal to 1/6, then minimum number of red balls in the bag is equal to?The Attempt at a Solution P =...
  20. T

    Minimum capacity rate ratio of a heat exchanger

    Homework Statement Here is the question: I am confused about the solution. In the solution they found that the minimum capacity rate ratio R = C_min/C_max was zero. This is because of the term I circled in the red which I do not understand why it goes towards infinity. Here is the solution...
  21. C

    Lagrangian Multipliers to find maximum and minimum values

    I'm just learning this theory and the maths is really trivial but the theory is slightly confusing me. I understand that if we have some function z=f(x,y) and we graph this on a three dimensional set of axis we will have some surface, we can then extend this by creating level curves in the...
  22. MexChemE

    Virial equation, minimum pressure point

    Good evening PF! I'm having trouble figuring out how to attack this problem. I have tried two different ways but I don't know if either of them is correct. Homework Statement Using the provided virial coefficients, determine analytically the pressure at which the graph of PV versus P for N2 at...
  23. G

    Find the minimum value of the quantity

    Homework Statement There are two cubes of mass m. They are initially tied by a string tightly. They are kept from joining into each other because of a spring (which in this tied up state has a compression of ε) of coefficient of stiffness k. Find the minimum value of the quantity εk/mg so...
  24. C

    MHB What are the values of b for a minimum of x^2 + bx - 25?

    Find the values of b such that the function has the given minimum value. f(x) = x^2 + bx - 25; Minimum value: - 34
  25. S

    Determine minimum value of integral

    Homework Statement Determine minimum value of integral ##B(y)=\int_{0}^{2}({y}')^2dx## for function ##y\in C^1(\mathbb{R})## and ##y(0)=y(2)=0## and ##\int_{0}^{2}y^2dx=4## Homework Equations The Attempt at a Solution IF I am not mistaken, the idea is to first find the function...
  26. L

    MHB Maximum and minimum 283: minimize cost

    283) you want to build a volume "V" shaped geometric body torque limited cylindrical half-spheres. If the material lower semisphere costs twice as much as the material of the sides, and the material of the upper hemisphere costs three times, calculate the dimensions of the body more economic...
  27. W

    Minimum Spanning Tree of a Graph Solutions

    I recently wrote a program that implements a slightly modified version of Prim's Algorithm to find a minimal spanning tree and it seems to work correctly. However, I am doubtful because my prof claims that this certain tree has more than 1 solution but my program gives only one solution. Note...
  28. T

    Minimum speed of the bullet to penetrate a sphere

    Homework Statement A bullet of mass m and charge q is fired towards a solid uniformly charged sphere of radius R and total charge + q. If it strikes the surface of sphere with speed u, find the minimum speed u so that it can penetrate through the sphere. (Neglect all resistance forces or...
  29. A

    Understanding the Minimum Energy Principle in Thermodynamics

    Confuses me. In which case is it the helmholtz free energy, the gibbs free energy, the energy that gets minimized and why? Also is it consistent with energy conservation and how is that possible if you use it on two systems exchanging energy. Can we know the total energy of the total system for...
  30. D

    Determine the minimum initial velocity

    Homework Statement Determine the minimum initial velocity V0 and the corresponding angle theta at which the ball must be kicked in order for it to just cross over the 3 m fence. Given: Horizontal distance=6m vertical distance=3m Homework Equations equation 1: Vy2=(V0)2+2a(y-y0)...
  31. L

    MHB (2) Find the height of the cone's lateral surface minimum confined to a sphere of RADIUS R.

    (2) Find the height of the cone's lateral surface minimum confined to a sphere of RADIUS R. the answer is (2 + sqrt(2)) R
  32. L

    MHB Sos help with maximum and minimum

    Between all the cones whose generatrix length given is L, determine the one with the highest volume? :confused: the answer is h= L/sqrt(3) and R = Sqrt(6)L/2
  33. M

    First-Order Extrema in Classical Mechanics , Theoretical Minimum

    First-Order Extrema in "Classical Mechanics", Theoretical Minimum In the 3rd lecture of Classical Mechanics, 2011, by Dr. Susskind in his Theoretical Minimum series, he talks about calculating extrema, saddle points, etc. to "first order". "if you move a little bit, the potential is zero, to...
  34. J

    MHB Find Minimum of PA + PB on $3x+2y+10=0$ given $(4,2)$ and $(2,4)$

    Find a point $P$ on the line $3x+2y+10=0$ such that $PA+PB$ is minimum given that $A$ is $(4,2)$ and $B$ is $(2,4)$ My Try: Let Coordinate of point $P$ be $(x,y)$. Then $PA = \sqrt{(x-4)^2+(y-2)^2}$ and $PB = \sqrt{(x-2)^2+(y-4)^2}$ Now Let $f(x,y) =...
  35. Saitama

    Finding the minimum frequency

    Homework Statement A heavy string is tied at one end to a movable support and to a light thread at the other end as shown in figure. The thread goes over a fixed pulley and supports a weight to produce a tension. The lowest frequency with which the heavy string resonates is 120 Hz. If the...
  36. 4

    Susskind's theoretical minimum

    If anyone out there has worked through Susskind's book, I have two questions on the Lagrangian to Hamiltonian section, any help would be greatly appreciated: 1) In Lecture 8 exercise 2, he wants you to calculate take the Lagrangian of L=1/2ω d/dt q - ω/2 q^2 as a Hamiltonian and says it...
  37. C

    Minimum distance between a disk in 3d space and a point above the disk

    Hi all, How can I calculate the minimum distance between the perimeter of a disk in 3d space and a point above the disk? (the point can be inside or outside the area above the disk) I've been trying to work this out for a while, but I'm getting no where. For example, a point at (1,1,1)...
  38. P

    Minimum force required to turn over box.

    Hi, Homework Statement I would like to determine the position, magnitude and direction of the minimum force F required to turn over the box shown in the attachment (width - B, height - H). The (uniform) box is placed on a horizontal plane. The coefficient of friction between the box and the...
  39. dexterdev

    What is the minimum temperature to make lasingaction possible in laser

    Hello friends, I would like to know the temperature dependence of LASER for its action. At What temperature the laser action is possible and realisable? Any references, links and help is appreciated.
  40. M

    Finding the minimum of a 2 variable function.

    Homework Statement ind the absolute minimum value of the function f(x,y) = 6 + 3xy -2x- 4y on the set D. D is bounded by the parabola y=x^2 and y=4 Homework Equations Partial derivatives and a theorem where if F is continuous on a closed bounded set in R(2) then F has an abs. max and...
  41. Q

    Minimum daily whole-body X-ray dose to cause radiation poisoning?

    Hi, If I were given a whole-body X-ray dose daily for many days (a year for example) What would be the highest dosage per day which would result in an increased cancer risk but no noticeable signs of radiation poisoning by the end of the year, or during? And also, I've read that a 2KRad dose...
  42. C

    Minimum Angular Velocity Problem

    First off, I want to apologize for posting more than one question. I just discovered this site, so I wanted to check my work while I am able to. Thank you again. Homework Statement A massless rope of length L = 1 m is swung in the vertical plane, with a bob of mass m = 1 kg attached to its...
  43. M

    Finding Minimum Force Applied to Keep Objects Stationary?

    The coefficient of static friction between the crate m1=3.00kg and the incline with θ=35° shown in the figure below is .300. What minimum force F must be applied to the crate perpendicular to the incline to prevent the crate from sliding down the incline? mgsinθ μsmgcosθ f=ma I don't quite get...
  44. D

    Why is there only one minimum deviation point for a prism?

    I'm having an issue in comprehending the minimum deviation offered by a prism. The fact that we could use the symmetry argument about the angle of incidence and angle of emergence being equal for minimum deviation make sense to me but I couldn't understand why we can be so sure of exactly 1 such...
  45. J

    Calculating minimum reserve factor in an I-beam

    Hi Everyone, I am confused about how to calculate the minimum reserve factor against yielding and fatigue in an I-beam clamped at both ends. I have calculated the maximum bending stress from it's deflection due to a load in the middle of the beam, and I know the cross-sectional area of the...
  46. anemone

    MHB What is the minimum value of |a|-|b| when $\log_4 (a+2b)+\log_4 (a-2b)=1$?

    If $\log_4 (a+2b)+\log_4 (a-2b)=1$, find the minimum of $|a|-|b|$.
  47. anemone

    MHB Finding the Minimum Value of a Complex Expression

    Determine the minimum value of $\left( \sqrt{x^2-8x+27-6\sqrt{2}}+ \sqrt{x^2-4x+7-2\sqrt{2}} \right)^4$ where $x$ is a real number.
  48. U

    What is the minimum length chord of a circle passing through a specific point?

    Homework Statement px+qy=40 is a chord of minimum length of the circle (x-10)^2 + (y-20)^2 = 729 . If the chord passes through (5,15), then p^{2013}+q^{2013} is equal to Homework Equations The Attempt at a Solution Let chord length be L \frac{L}{2} = 729-...
  49. D

    Minimum work and work-energy theorem

    Hi guys, I've got a doubt concerning to the minimum mechanical work and the work-energy theorem. Consider the following Tippens' problem (8.4): A 5-kg hammer is lifted to a height of 3 m. What is the minimum required work? The answer looks very simple and inocent, W = weight times...
  50. N

    Minimum Rate of Rocket Fuel Consumption

    Homework Statement A rocket which has an exhaust speed of 8.0 km/s is to have a vertical acceleration of 8.0 m/s2. If the initial mass of the rocket is 6.0x106 kg, at what minimum rate must the rocket consume fuel? Homework Equations F = ma F = RVREL The Attempt at a Solution F...
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