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.

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

    Minimum separation between pucks

    Two frictionless pucks are placed on a level surface 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 of puck 2...
  2. DryRun

    Derivative of f(x) to find its maximum and minimum values

    Homework Statement If λ is a positive constant, determine the maximum and minimum values of f(x) = 9(4-3x^2)(λ-λ^-1-x) and show that the difference between them is 4(λ+λ^-1)^3. Find the least value of this difference as the parameter λ is varied. The attempt at a solution I expanded the...
  3. G

    Minimum angular separation for viewing stars

    Homework Statement What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? Homework Equations Rayleigh criterion: θ=(1.22*λ)/D The Attempt at a Solution This problem doesn't give very much to go on so I think...
  4. F

    How to find the minimum of an integral with calculus of variations

    I need to find the minimum of this integral F=∫ (αy^-1+βy^3+δxy)dx where α, β and δ are constant; y is a function of x the integral is calculated over the interval [0,L], where L is constant I need to find the function y that minimizes the above-mentioned integral The integral is subject to...
  5. G

    Minimum energy of an electron trapped in a nucleus

    We were asked to estimate the minimum kinetic energy of an electron trapped within a nucleus having diameter d . My solution was this: find the momentum of the electron (via de Broglie relation) and use a very standard kinetic energy formula, like this (assuming minimum energy state has a...
  6. D

    How to determine maximum and minimum for Lagrange Multiplier?

    Homework Statement Find the minimum and maximum values of the function subject to the given constraint f(x,y) = x^2 + y^2, 2x + 3y = 6 Homework Equations \nablaf, \nablag The Attempt at a Solution After doing all the calculation, x value and y value came out to be...
  7. Z

    Find a function with the same maximum and minimum? What?

    Homework Statement Name two different functions that share a maximum and minimum at -4. Homework Equations The Attempt at a Solution This is a vague question. I guess horizontal lines would be one. I can't think of another. A function that is always a point? Thanks.
  8. K

    Cylindrical Container Minimum Cost

    Homework Statement An industry is to construct a cylindrical container of fixed volume V, which is open at the top of. Find the quotient of the height by radius of the base of the container so that manufacturing cost can be minimal, given that the unit area of the base costs (b) twice the...
  9. J

    Finding the local minimum of a graph

    Homework Statement The question provides a graph and asks for the local minimums. I attached a picture with the graph. 2. The attempt at a solution I said the local minima are when x=0,2,5. However the answer key suggests they are at 1,2,5. Could someone please explain why 1 is a...
  10. N

    Local Minimum of y=x(e^1/x): Why Not (1,e)?

    I found that the local minimum of y=x(e^1/x) is (1,e) but can someone tell me why the absolute minimum is also not (1,e) and can someone tell me why this function does not have an absolute minimum? Thanks
  11. J

    What is the Minimum Deviation Angle of a Prism with an Apex Angle of 50 Degrees?

    Homework Statement The apex angle of a prism is 50.0^\circ and the index of refraction of the prism material is 1.66. What is the minimum deviation angle? Homework Equations Snell's Law: n_1\sin(\theta_1)=n_2\sin(\theta_2)The Attempt at a Solution How would you approach this? I get a...
  12. S

    Minimum Coefficient of Friction calculation using Centripetal force

    A() 61 kg child stands at the rim of a merrygo-round of radius 1.05 m, rotating with an angular speed of 1.87 rad/s The acceleration of gravity is 9.8 m/s^2 What is the child’s centripetal acceleration? Answer in units of m/s What minimum force between her feet and the floor of the...
  13. Delta2

    Photoelectric effect minimum frequency

    Assume fmin is the minimum frequency of photon required for the electron to be ejected. Why can't we have an electron absorbing 2 photons of frequency fmin/2 thus the total energy will be 2h*(fmin/2)=h*fmin thus the electron to be ejected?
  14. M

    Finding the Optimal Angle for Maximum Projectile Distance

    What is the minimum speed the rock must have, when it leaves the sling to travel exactly 400 m? The only equation I can think of using is... y = (tan(theta initial))x - (g/(2Vx0^2)x^2 y = 0 x = 400 However I have to many unknowns... I need to solve for V but I don't know Vx0 or Vy0 or...
  15. L

    Tricky minimum distance vector problem

    Homework Statement The line intersects a plane at an angle alpha=2pi/3. The line is defined by r=μn, and the plane by r.m=0, with n and m unit vectors. Calculate the shortest distance from the plane to the point on the line with μ=2. Homework Equations ? The Attempt at a Solution...
  16. R

    Peltier minimum temparature to work

    Dear Experts, I like to check with you, is there a minimum temperature for a peltier thermal couple to start generating electricity? I tried putting a peltier on a stove which i was using to cook some food. There is no fire on the stove but its very hot. I place the hot side of the...
  17. G

    Lagrange Multipliers: Minimum and Maximum Values

    Homework Statement I am trying to find the min and max values of f(x,y)=2x^2 + 3y^2 subject to xy=5. Homework Equations f(x,y)=2x^2 + 3y^2 subject to xy=5 \mathbf\nablaf=(4x, 6y) \mathbf\nablag=(y,x) The Attempt at a Solution When I go through the calculations, I end up with two critical...
  18. O

    Asteroid falling to earth, what will be minimum speed and KE on planet?

    Homework Statement The gravitational pull of the Earth on an object is inversely proportional to the square of the distance of the object from the center of the earth. At the Earth's surface this force is equal to the object's normal weight mg, where g = 9.8m/s^2, and at large distances, the...
  19. T

    Find minimum energy to escape an orbit

    Homework Statement When it orbited the Moon, the Apollo 11 spacecraft ’s mass was 7170 kg, its period was 121 min, and its mean distance from the Moon’s center was 1.87167 × 106 m. Assume its orbit was circular and the Moon to be a uniform sphere of mass 7.36 ×1022 kg. Its orbital speed was...
  20. M

    Minimum number of edges in a graph of order n with chromatic number k

    What is the minimum number of edges in a k-chromatic connected graph of order n? I have read somewhere that this number is equal to \left(\begin{array}{c} k \\2 \end{array} \right) +n -k I was thinking about using K_k, the complete graph with k vertices. We know that the number of edges...
  21. A

    Solve Physics Homework with Minimum Speed | Q4 Equations | Physics Forums

    Homework Statement See Q4 https://www.physicsforums.com/attachment.php?attachmentid=39736&d=1318081996" Homework Equations The Attempt at a Solution I am not sure whether my direction of thinking is correct. I have a bad feeling about my work. Is my work correct...
  22. W

    Wave packets & Fourier analysis: minimum pulse duration w/ given frequency

    Homework Statement I: A telephone line can transmit a range of frequencies \Delta f = 2500 Hz. Roughly what is the duration of the shortest pulse that can be sent over this line? II: A space probe sends a picture containing 500 by 500 elements, each containing a brightness scale with 256...
  23. E

    Mathematica Minimum function path on a surface with Mathematica

    Hello, I am trying to solve the following problem using Mathematica: I have (defined) a surface of two variables, say f(a,b). Now I want to plot a path (a line) that corresponds to min[f(a,b)] starting from certain valus of a and b (say a0,b0). In other words the path always proceeds...
  24. K

    Find a curve such that time travel is minimum

    Homework Statement [PLAIN]http://vctson.com/temp/curve.jpeg Find the curve y = f(x) so that the ball travels from A to B in minimum time? Thanks!
  25. S

    Minimum Time for Elevator to Reach Level B?

    Homework Statement An elevator of mass (m), is pulled upwards by a massless rope, which can bear a maximum tension of nmg (n-any natural number > 1). If the elevator starts from rest at point A, and it rises to level B where it comes to rest again.(momentary rest is sufficient). Find the...
  26. S

    Exam : find the minimum value of this function and constrant

    Homework Statement Given x+y = 6\sqrt2 a + b = 6\sqrt 2 Find minimum value of \sqrt{x^2 + a^2} + \sqrt{y^2+b^2} 2. The attempt at a solution I had plotted the f(x,y) with constant a ,b And that's all. I don't realize how to do this. Any comment will be appreciated
  27. T

    Finding minimum distance to stop using friction and velocity

    Homework Statement A car is traveling at 44.4 mi/h on a horizontal highway. The acceleration of gravity is 9.8 m/s2 . If the coefficient of friction between road and tires on a rainy day is 0.13, what is the minimum distance in which the car will stop? (1 mi = 1.609) Answer in units of...
  28. P

    Finding the maximum or minimum value of a funtion to the nth degree?

    How do i find the maximum or minimum value of a function to the nth degree? We've only done this for a quadratic function/equation in my class, but how would u find this for a higher degree such as x^3, x^4 or x^n? Does it follow the pattern of a power function in that x^4 is similar to x^2?
  29. J

    Let f(x)=3x^2+6x-10For which input x is the value of the function a minimum?

    And what is the minimum value? ( -b/2a is for a maximum problem, but how do you get through this kind of problem?)
  30. P

    Polarization: minimum thickness for quater-wave plate

    a certain bifringement material has indexes of refraction n1 and n2 for the two perpendicular components of linearly polarized light passing through it. the corresponding wavelengths are λ1=λ0/n1 and λ2=λ0/n2, where λ0 is wavelength in vacuum. If crystal is to function as a quarter-wave plate...
  31. D

    How Do You Calculate the Minimum Radius of Curvature for a Banked Roadway?

    Homework Statement A civil engineer is asked to design a curved section of roadway that meets the following conditions: With ice on the road, when the coefficient of static friction between the road and rubber is 0.1, a car at rest must not slide into the ditch and a car traveling less than 80...
  32. C

    What is the bare Minimum Lagrangian\Hamiltonian mechanics

    I am currently trying to self learn general relativity. I understand you need knowledge of an action principle, but what I am not so sure of is how deep of an understanding I need. I currently have a book on classical field theory and I am at the point of basic Lagrangian's and holonomic...
  33. P

    Polarization, minimum thickness of a quarter wave plate

    Homework Statement show that minimum thickness of quarter wave plate is d=\lambda0/4(n1-n2 Homework Equations \lambda1=\lambda0/n1 \lambda2=\lambda0/n2 The Attempt at a Solution for quater wave plates wavelengths must differ by 1/4 i can see its simple but i just can't see what...
  34. F

    Banked road (find the minimum radius)

    Homework Statement I've already answered the first part. The second part is what's giving me trouble. Homework Equations Here's how I solved #10: Since the car is at rest, we're only worried about the force of friction and the parallel component of gravity: F_f = F_g *...
  35. B

    How to Calculate Minimum Duct Thickness for Vacuum at 150°C?

    How to calculate the minimum thickness of a duct which hold vacuum inside at 150 deg c? And what's the basic to select the stiffer size and distance between each stiffener. Thanks.
  36. P

    How Is Minimum Sensitivity Calculated in Instruments?

    Homework Statement The attempt at a solution First, we define sensitivity as the change of an instrument or transducer's output per unit change in the measured quantity. So to find the minimum sensitivity I simply plugged in the largest possible input (5.0 mm) into the equation: E =...
  37. A

    Optimizing astable multivibrator for minimum power consumption

    In short, if I want a given frequency, and I use BJT transistors, what would be the optimum capacitors and resistors in a multivibrator. Example: http://upload.wikimedia.org/wikipedi...tivibrator.svg So, what are the optimum values C1=C2, R1=R4, and R2=R3?
  38. G

    Deriving minimum escape height and velocity

    Homework Statement At which minimum height atoms or molecules are able to leave Earth’s atmosphere to space? To obtain an approximate estimate use the assumption that molecules can leave if the mean free path, λ, is larger than the scale height H. Write λ and H as a function of height and...
  39. S

    What Minimum Height Fish Must Spot Pelican to Escape?

    Homework Statement If it takes a fish 0.19s to perform evasive action, at what minimum height must it spot the pelican to escape? [Suppose a pelican starts its dive from a height of 15.9 m and cannot change its path once committed] If it takes a fish 0.19s to perform evasive action, at...
  40. R

    Probability- minimum inventory to satisfy demand

    Homework Statement There are 20 customer locations, the demand in each location is normal with mean 60 and SD 20. All 20 locations have independent probabilities. The goal is to cover all of the demand in a month at least 95% of the times. What's the minimum total inventory the company should...
  41. A

    Minimum Critical Power Ratio (MCPR) for a BWR

    From Todreas & Kazimi, Nuclear Systems I, Problem 2-3 1. Homework Statement Calculate the minimum critical power ratio for a typical 1000 MWe BWR operating at 100% power using the data in Tables 1 – 2, 1 – 3, and 2 – 3. Assume that: a) The axial linear power shape can be expressed as...
  42. D

    Calculating minimum time intercept for two accelerating craft

    Given two spacecraft with different accelerations and vectors, how would I go about calculating the minimum time solution for one spacecraft to intercept the other, assuming that the interceptee will maintain its current acceleration and vector? Note that the interceptor is not attempting...
  43. L

    IRF3000 Minimum and Maximum Voltage

    Greetings! I am actually confused on what is the maximum and minimum voltage at the drain pin of IRF3000 MOSFET. Here is the datasheet: http://www.datasheetcatalog.org/datasheet/irf/irf3000.pdf Here is what the datasheet says: VDSS = 300V I wonder if this is the maximum or minimum voltage. I...
  44. A

    What Is the Minimum Launch Angle to Hit a Target 200ft Away?

    I wasn't able to figure this problem out, so I looked at the answer. The answer is 19.06 degrees and 2.116 seconds. Using these values in the kinematic equations, I calculated that at x=200 ft, the height of the projectile is y=50ft. When x=200ft, shouldn't y=0ft? I thought the problem is asking...
  45. 5

    Wire cut to shape circle and square so total area is maximum or minimum

    Homework Statement A piece of wire 40 cm long is cut into two pieces. One piece is bent into the shape of a square and the other is bent into the shape of a circle. How should the wire be cut so that the total area enclosed is a (a) maximum and (b) minimum? Homework Equations Area...
  46. S

    What Is the Minimum Power Required for a Heat Pump to Maintain 20°C?

    Hello, Just wondering if you could help me out with this question; A house is to be maintained at 20°C in winter and the heating load is 10kW. The outside temperature is -5°C. If a heat pump is to be used to heat the house, what is the minimum power required to drive the heat pump...
  47. M

    Minimum Power of the engine on a car climbing a hill

    Hi, Can someone please solve this question for me, including step by step instructions ? I can solve the ones where the mass of a car is given, but can't do this one. Homework Statement Setting off from rest the car accelerates uniformly and climbs a steep hill which has a gradient of 1...
  48. J

    Incredibly close to a modular arithmetic proof by minimum counterexample

    As stated in the title, I am trying to prove a statement by minimum counterexample involving modular arithmetic. My problem is producing the contradiction, but I feel so close! (The contradiction is p^m | (1 + p)^{p^{m - 1}} - 1) Homework Statement Let p be an odd prime and let n be a...
  49. S

    Minimum Absolute Value of a nxn Matrix Determinant

    The following matrix problem occurred to me. I figured out the answer and would like to pose the problem. It's easy but would be best for an undergrad math major. The question: Consider a square n by n matrix with entries 1, 2, ..., n squared. Find a way to arrange these entries so that the...
  50. J

    Laser focusing, minimum beam width

    I need to know the beam width of a pulsed yag laser (280.5 nm) when focused, assuming its completely guassian. The laser is focused through a bi-convex lens with f=15cm. I've found expressions for w(z), but they are all in terms of the minimum waist, w0, which I don't have and is probably the...
Back
Top