Minimum Definition and 1000 Threads

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. 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...
  2. M

    Minimum coefficient of static friction problem

    Homework Statement I know the diameter is 30 m and the radius is 15 The period for one revolution I suppose is some multiple of 7.03 Homework Equations The Attempt at a Solution I couldn't think of an equation where friction was involved where rotational motion was concerned. I...
  3. N

    Minimum Energy Curve Derivation Help

    I've been trying to find the equation of the line of no strain of a bent linearly elastic beam of fixed length. (in a plane, given just its endpoints, or both the end points and the slopes at the endpoints) I imagine this problem has been solved many times, but I couldn't find a solution online...
  4. U

    Minimum linear velocity attained by sphere

    Homework Statement A sphere of mass M and radius R is moving on a rough fixed surface, having co-efficient of friction μ, with a velocity v towards right and angular velocity ω clockwise. It will attain a minimum linear velocity at time (take v>ωR) The Attempt at a Solution Since v>ωR the...
  5. B

    Minimum Force (roller over step)

    On an inclined plane (φ = 12°), calculate the minimum force P required to pull the roller over the step. The radius of the roller, r = 0.685m My attempt The sum of the Moment force about the system = 0 0 = mg(sin(φ)*(r-h) + mg(cos(φ)*√[r2-(r-h)2] - P(cos(θ)*r-h) P(cos(θ)*r-h) =...
  6. H

    Finding minimum for an equation with two variables

    Homework Statement I have the equation: x^2 + 2*x*y + 5*y^2 - 4*x - 6*y +7 and I have to find the minimum value I'm getting something that looks half like the correct answer, but not quite right... Homework Equations The answer from the answer book is: [x + 2*(y - 1)]^2 + (y + 1)^2...
  7. B

    Finding Minimum and Maximum values.

    Homework Statement Find the minimum and maximum values of x and the values of f(x) at these points if: f(x) = 10.9sin(0.7x)+2.1cos(0.7x) The range is (-.27,8.7) Homework Equations I don't know/think there are any The Attempt at a Solution I found f'(x) =...
  8. C

    Find the minimum speed of a particle

    The position function of a particle is given by r(t)=⟨−5t2,−4t,t2+1t⟩. At what time is the speed minimum? I found the speed function by taking the magnitude of the velocity which is lv(t)l = sqrt(108t^2 +4t + 17) I then took the derivative of the speed to find the critical point...
  9. W

    What Determines the Minimum Value in a Quadratic Equation?

    Homework Statement 4. If x1, x2 are two real number roots of real number coefficient quadratic equation: $$x^2 -2mx + m + 2 =0$$ Solve the following questions: (1) What are the values of m so that x1=x2? (2) What are the...
  10. P

    Maximum and Minimum Kinetic Energies following decay

    Homework Statement An unstable particle of mass M = m1 + m2 decays into two particles of masses m1 and m2, releasing an amount of energy Q. Determine the kinetic energies of the two particles in the CM frame. Given that m1/m2 = 4, Q = 1 MeV, and that the unstable particle is moving in...
  11. TheFerruccio

    Principal of Minimum Complementary Energy

    Homework Statement I am reading through this link: http://en.wikiversity.org/wiki/Introduction_to_Elasticity/Principle_of_minimum_complementary_energy I need to show that a certain thing follows from a previous thing. I will elaborate in the next section. Homework Equations I am stuck...
  12. U

    What is the minimum value of 'b' in the given equations?

    Homework Statement Let a,b,c be in G.P. and a-b,c-a,b-c in H.P. If both roots of (a+c)x^2 + bx + 4b^2=0 are positive and minimum value of 'b' be k then value of |[k]| Homework Equations The Attempt at a Solution Let a,b,c be denoted by a,ar,ar^2. now \frac{2}{c-a}=\frac{1}{a-b}+\frac{1}{b-c}...
  13. M

    Minimum and maximum values of the objective quantity?

    Hi, I am not sure how to solve the following high school algebra problem. As shown in the attached file, I have plotted the first three equations, but I don't know what to do with the fourth equation. Find the minimum and maximum values of the objective quantity. 2X + Y ≤ -8 3X - 2Y...
  14. U

    What Is the Minimum Number of Real Roots for the Given Equation?

    Homework Statement Let f(x) be a non-constant twice differentiable function defined on R such that f(x)=f(1-x) and f'(1/4) =0 then what is the minimum number of real roots of the equation (f"(x))^2+f'(x)f'''(x)=0. The Attempt at a Solution f'(x)=-f'(1-x) f"(x)=f"(1-x)...
  15. C

    Calculating Minimum Plate Thickness

    I'm working on a project where I know that force will be applied by a bolt pulled through a hole in the center of an aluminum plate for load testing purposes (2200 lbs, actual worst case max force around 1500 with a 1.5 times safety factor), the dimensions (27"x6.25"x?), and the material (5052...
  16. T

    Find minimum force to move block

    Homework Statement "Figure 1 shows two wedge blocks A and B, weighing 1000 N and 50 N, respectively, in contact. The coefficient of friction between all contacting surfaces is 0.1 . What would be the smallest value of P required to push the block A upwards? What would be its value if there...
  17. M

    Minimum area of a shoe heel exerting pressure

    Homework Statement The floor on a dance hall can take a pressure of 3.5 MPa before getting damaged. A lady of mass 60kg wears high heeled shoes. What is the smallest area of the sole of a heel she could wear such that it won't permanently damage the floor? Homework Equations Area =...
  18. 5

    Minimum value on an open continuous function

    Homework Statement Suppose that f is a continuous function on (a,b) and lim_{x \rightarrow a^{+}} f(x) = lim_{x \rightarrow b^{-}} f(x) = \infty. prove that f has a minimum on all of (a,b) The Attempt at a Solution I have not tried an actual attempt yet. The only think I can...
  19. B

    What is the minimum tension (in Newtons) in the angled rope?

    Homework Statement A hanging 2.00-kg (19.6N) block begins to accelerate a 4.28-kg (41.94N) block on a horizontal surface. But a rope connected to the wall at 26.9 degrees brings the block to an equilibrium position. The coefficient of friction between the surface and the 4.28-kg (41.94N) block...
  20. anemone

    MHB How can I find the minimum value of xy given certain conditions?

    Hi MHB, I've found an interesting and difficult (at least for me) problem which asks us to find the minimum value of $xy$ and no matter how serious that I tried, I couldn't get myself out of the misery of finding what the question wanted. Could anyone help? Problem: Find the minimum...
  21. T

    Determine the magnitude of the minimum acceleration

    Homework Statement A 76-kg petty thief wants to escape from a third-story jail window. Unfortunately, a makeshift rope made of sheets tied together can support a mass of only 58 kg.Determine the magnitude of the minimum acceleration at which the thief can descend using the rope. Homework...
  22. D

    Finding the Minimum θ for Rock Thrown at Speed v_0

    Homework Statement A boy throws a rock at speed v_0 at angle α from a balcony of height h. When the ball hits the ground, its velocity makes an angle θ with the ground. What is the minimum value of θ in terms of h and v_0 Homework Equations basic kinematics equations. The...
  23. MidnightKat

    A Theoretical Minimum | Looking for Guidance

    Hey everyone! I am on a quest to understand the world in which we live in better. In doing so I'm making a stop at Physics. I was suggested the book "A Theoretical Minimum - What you Need to Know to Start Physics". I am in love with this book and cannot put it down. At the moment I am...
  24. MarkFL

    MHB Find the altitude of the triangle with minimum area

    Consider the isosceles triangle circumscribed about the ellipse: Find the altitude of the triangle when its area is minimized.
  25. J

    Good follow up to The theoretical Minimum?

    I'm currently self teaching myself physics and I have never seen physics before this. The book that I'm using is The Theoretical Minimum - What You Need to Know to Start Doing Physics by Leonard Susskind. I plan to be finished with the book by mid next week and my question is what's a good...
  26. anemone

    MHB What is the AM-GM Inequality Used For?

    Determine the minimum value of \frac{\sec^4 a}{\tan^2 b}+\frac{\sec^4 b}{\tan^2 a} over all $a,b \ne \frac{k \pi}{2}$ where $k$ is in $Z$.
  27. anemone

    MHB Find the Minimum Value of |x|-|y|

    If \log_4(x+2y)+\log_4(x-2y)=1, find the minimum value of $|x|-|y|$.
  28. T

    Finding minimum values for problems

    Homework Statement Hi guys, really stuck on this one. A 1000N black is to be lifted using a chain sling (as shown). If the tension in the chain sling is not to exceed 1300N, find the shortest sling ABC that can be used. A is the most left point, B is the most right point and C is the top...
  29. A

    Optimizing Mirror Widths for Periscope Design

    Homework Statement A periscope is made of two good mirrors. Find the ratio of the minimum widths of the mirrors if the distance between them is L and the distance from the lower mirror to the eye of the observer is l.The objects viewed through the periscope are at a great distance from it...
  30. B

    Minimum distance around a track

    Homework Statement A motorbike starts from rest at time t=0 and begins to accelerate around a circular track of radius 10m. Eventually, at time t=t1 the motorbike reaches the maximum velocity possible without slipping off the track. What's the minimum length in meters the motorbike must travel...
  31. P

    Sphere has the minimum surface area?

    How do you prove that for a given volume, sphere has the minimum surface area?
  32. S

    MHB How to Graphically Determine the Minimum of a Function Under a Constraint?

    Problem: The sum of two real numbers is 1. What is the minimum value of the sum of the squares of the two numbers? I have already managed to solve the problem algebraically (by substitution and completing-the-square we arrive at a minimum value of 0.5), but what I am interested in is a...
  33. J

    Find Max & Min Pipe Diameters for Combined Area

    Can someone help me as a am racking my brain. I have been given this question; Two oil pipes with circular cross-sections are subject to the constraint that the sum of their diameters is 1m. By using careful reasoning find the diameters that give the maximum and minimum possible combined...
  34. sunrah

    Estimate minimum proton lifetime from internal radiation rate

    Homework Statement Estimate the minimum proton lifetime from the fact that you haven't yet died from internal radiation. Use the following conditions: 1.) At a dosis of Dmax = 2 Greys per year (Jkg-1 yr-1) you would already be dead. 2.) The human body is made mostly of water. 3.) The total...
  35. H

    Minimum acceleration needed for box to not fall from accelerating cart

    Homework Statement You place a 2kg box against the side of a cart. Nothing is used to stick the box to the cart which accelerates to the right. The coefficient of static friction is 0.6. Find the minimum acceleration of the cart needed so that the box does not fall Homework Equations...
  36. Saitama

    No problem! Glad I could help :)

    Homework Statement Let x, y, z be positive real numbers such that ##x+y+z=1##. Determine the minimum value of \frac{1}{x}+\frac{4}{y}+\frac{9}{z} Homework Equations Cauchy-Schwarz inequality maybe? The Attempt at a Solution Applying the Cauchy Schwarz inequality...
  37. X

    Finding minimum electrostatic energy

    Homework Statement There are 2 small metal balls of radius r1 and r2 and are kept at very large distance , what should be ratio of charges on them for electrostatic energy to be minimum Homework Equations E=Q^2/(8πϵR) Self Energy The Attempt at a Solution In this problem, electric...
  38. S

    Finding the Minimum Mean Square Estimator for Scalar Parameter w

    I am not able to understand how to go about this problem: Find the minimum mean square estimator for the scalar parameter w based on the scalar observation z = ln w + n where f(w) =1 if 0<=w<=1; 0 else: and f(n) =e^-n if n>= 0; 0 else I did f(z/w) = (f(n))...
  39. S

    MHB Linear Minimum Mean Square Estimator

    I am not able to understand how to go about this problem: Find the minimum mean square estimator for the scalar parameter w based on the scalar observation z = ln w + n where f(w) =1 if 0<=w<=1; 0 else: and f(n) =e^-n if n>= 0; 0 else I did f(z/w) = (f(n))/ g'(n)...
  40. Fernando Revilla

    MHB Minimum speed (~V3N0M~'s question at Yahoo Answers)

    Here is the question: Here is a link to the question: The position function of a particle is given by r(t)=<−3t^2, −2t, t^2+4t>? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  41. J

    What is the minimum observer energy?

    Question: Suppose I want to measure the polarity of a photon. What is the minimum amount of energy I need to invest? What are the needed forms? (Examples might include rest mass for storage devices, energy in interacting particles, etc.) I'm looking for an ideal device. Background: I have a...
  42. JK423

    Is there a minimum mass for black holes?

    Hi all, I was (superficially) reading about the information loss paradox. Of what i understood it's based on the complete evaporation of the black hole via hawking radiation, so in some sense all the energy of a black hole will eventually become radiation. The following question immediately...
  43. Saitama

    Thin film interference - Minimum thickness

    Homework Statement A lens (##\mu##=1.5) is coated with a thin film of refractive index 1.2 in order to reduce the reflection from the surface at ##\lambda=480 nm##. Find the minimum thickness of the film which will minimize the intensity of reflected. [Assume near normal incidence]Homework...
  44. A

    Minimum thickness n/wavelength equation?

    Homework Statement White light is incident normally on a thin soap film (n = 1.33) suspended in air. 1. What is the minimum thickness of the soap film that will constructively reflect light of wavelength 477 nm? 2. What is the next minimum thickness of the soap film that will...
  45. S

    MHB Fraction between 96/35 & 97/36 with minimum denominator

    Q: Find a fraction between \frac {96}{35} and \frac {97}{36} that has the smallest possible denominator. My thoughts: Let's assume this fraction is of the form \frac pq where \gcd(p,q) = 1. Since \frac{96}{35} > \frac{97}{36}, we can write \frac{97}{36} < \frac {p}{q} < \frac{96}{35} \, . By...
  46. M

    Minimum Velocity to become airbourne

    I don't have a specific question, but was wondering how I would go about solving something like this: How would you calculate the minimum velocity that a bike, for example, would require to become airbourne off a circular ramp, if you are given say the radius of the ramp. Would you need more...
  47. J

    MHB Calculate pay off debt with minimum payment only?

    How can i calculate how many months it would take me to pay off a debt if I only make the minimum payment? let say I ow 5000 my interest rate is 6.4% my payment rate is 2% so If i calculate my minimum payment I know it be 100 more or less but I i just pay that minimum how many month it would...
  48. T

    Collisional excitation minimum velocity

    Homework Statement A sodium atom emits a photon with wavelength 818 nm shortly after being struck by an electron. What minimum speed did the electron have before the collision? Homework Equations E_{particle}\geq\Delta E_{atom} The Attempt at a Solution For the minimum energy...
  49. M

    Finding Minimum Thickness Problem

    A polymer coating has a n=1.30 and a base of n=1.50. What is the minimum thickness of the coating to reflect 500 nm least strongly?
  50. M

    Optimizing Polymer Coating Thickness for Reflectivity at 500 nm Wavelength?

    Homework Statement A polymer coating has a n=1.30 and a base of n=1.50. What is the minimum thickness of the coating to reflect 500 nm least strongly? Homework Equations The Attempt at a Solution
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