What is Minima: Definition and 113 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. D

    What is the difference between global and local maxima and minima?

    hi pf, this is my 1st math question. i am a undergraduate grade 12 student and my question is what is the basic difference between local maxima minima & global maxima minima?
  2. L

    Advanced Lagrange Maxima Minima Problems

    My first post and I am new to forum etiquette, please go easy on me. I have an exam in a few days and have a good grasp on the content. Historically the professor has been putting harder Lagrange questions on the exam than are available in my textbook and supplement books or that I can find...
  3. J

    Maxima and minima word problem

    Homework Statement A corridor of with a is at right angles to a second corridor of width b. A long, thin, heavy rod is to be pushed along the floor from the first corridor into the second. What is the length of the longest rod that can get around the corner? Homework Equations I...
  4. O

    Function with many local minima

    I've been working on a decentralized algorithm for finding local minima. Can anyone give me a few examples of mappings of the form F:R→R that have multiple local minima. I'm having problems defining neighbourhood on mappings from R2→R, so I thought I'll test it out on single variable functions...
  5. M

    Maxima and minima of functions of two variables

    Homework Statement Locate all relative maxima, relative minima ,and saddle points if any. f(x,y)=ysinx fx(x,y)=ycosx fy(x,y)=sinx ycosx=0 sinx=0 y=0 x=0,∏,2∏... up until infinity Critical points at (0,0),(∏,0),(2∏,0)... fxx(x,y)=-ysinx...
  6. J

    Maxima and minima of function

    Homework Statement Consider the following function of the variable r, r>/=0 y(r)=(r^(2)-5.50r)exp(-r) Find the value of the first derivative dy/dr at r=5.50 Homework Equations How do I solve this? I know its a simple derivative equation, but I can't seem to get it. I tried finding the...
  7. B

    Minima and maxima of a cycloid

    The cycloid is defined by the parametric equations x = a(t-sin(t)) and y = a(1-cos(t)) I am trying to find the set of points of relative extrema of a cycloid. I differentiated first to get \frac{dx}{dt} = a(1-cos(t)) and \frac{dy}{dt} = a*sin(t) Then, by the chain rule...
  8. S

    Finding the maxima and minima of a surface defined by datapoints

    Here's the problem I'm trying to solve, We have an N dimensional surface. We do not know the form of this surface however we have datapoints which are likely to be close, though not guaranteed to be on the surface, with some outliers. What I want to do is determine where in N dimensional...
  9. N

    Minima of sin function with domain (a, b)

    Homework Statement Consider the function f(x) = sin (x) in the interval x ∈ (p/4, 7p/4). The number and location(s) of the local minima of this function are? Homework Equations The Attempt at a Solution I have managed to get one of the minimas (i.e. 3p/2), however i am unsure about...
  10. R

    Is there a way to prove that A(x) is minimum when x=2r?

    The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle. is this way correct? assume that x is side of square and r is radius of circle k=4x+2∏r sum of...
  11. T

    Sound, phase difference and number of minima

    Homework Statement http://www.xtremepapers.com/papers/CIE/Cambridge%20International%20A%20and%20AS%20Level/Physics%20%289702%29/9702_s09_qp_2.pdf number 5b The speed of sound is 330ms-1 The frequency of sound from S1 and S2 is increased. Determine the number of minima that will be...
  12. N

    Maxima minima of function of several variables

    Could you pls advise me any reference (textbook for example) where I can find how to calculate maxima minima of function of several variables... or drop a couple of lines here.. Thank you
  13. D

    Calculus 1: Finding the maxima and minima of the function 1-x^(2/3)

    Homework Statement I am given the problem "A function is defined by f(x) = 1-x^(2/3). a. find the minima and maxima of the function. State whether they are relative or absolute. b. graph the function." Homework Equations I found the derivative and set it equal to zero. y' =...
  14. J

    What are the local maxima and minima of F(x)=(x^2)/(x+1)?

    Homework Statement F(x)=(x^2)/(x+1) Find critical points Find local maxima & minima Homework Equations None The Attempt at a Solution F'(x) = x(x+2)/(x+1)^2 crit points: -2,0,-1 f(-2) = -4 f(0) = 0 f(-1)=undef My book is telling me that f(0) is the minima, and f(-2) is the...
  15. K

    Find All local maxima and minima and all saddle points of the function

    Homework Statement ## f\left( x,y\right) =x^{2}-4xy+6x-8y+2y^{2}+10 ## ## f_{x}=2x-4y+6=0 ## ## f_{y}=-4x-8+4y=0 ## ## f_{y}=-4\left( x-y+2\right) ## -2=x-y, then solving fx and using this equality ## f_{x}=0=2\left( x-2y+3\right) =0 ## 2(-2 -y+3)=0 2(1-y)=0 y=1, then pluggin...
  16. E

    Concavity, inflection point, maxima, minima

    1.Find the vertical and horizontal asymptotes (if any), intervals of increase and decrease, local maximum and minimum values, intervals of concavity and inflection points for f(x) = 1 / (L - x^1/2) 3. I found the vertical asymptote to be "x=0" , I found the horizontal asymptote to be "y=0" So...
  17. Saitama

    Minima of 9tan2x+4cot2x | 65 Characters

    Homework Statement This is question is from my test paper. Find the minimum value of 9tan2x+4cot2x. I wasn't able to solve it during my examination. But later i remembered that i can find its minima which unfortunately my teacher still hasn't taught. :( I have a very little knowledge of it so...
  18. N

    Why the minima of potential of classical Lagrangian called ''vacuum expectation''?

    Please teach me this: Why the minima of potential of classical Lagrangian is called the ''vacuum expectation value of Phi(field function)''.Is it really a vacuum expectation value of field operator at the vacuum states(at this state,the potential part of classical Lagrangian equals zero)...
  19. R

    Optimization problem, local minima and feasible set

    Homework Statement minimiza f(x) = x_1 subject to (x-1)^2+y^2=1 (x+1)^2+y^2=1 Graph the feasible set, Are there any local minimizers and global minimizers? Homework Equations I have graphed the feasible set...
  20. F

    A problem in minima and maxima applicatios ( )please

    a problem in minima and maxima applicatios (urgent !)please Homework Statement A straight river 3km wide runs due east. A cable is to be laid from a point A on the south bank of the river to a point B on its north bank, 9km downstream from A. The cable is to be laid in a straight line from...
  21. Telemachus

    Proof of Saddle Roof at (0,0) for f(x,y)=(y-3x^2)(y-x^2)

    Hi there. I've got this function f(x,y)=(y-3x^2)(y-x^2), and I have to analyze what happens at (0,0) in terms of maxims and minims. But what I actually have to proof is that there's a saddle roof at that point. Theres is a critical point at (0,0). Let's see...
  22. A

    Michelson interferometer - maxima and minima

    URGENT! michelson interferometer - maxima and minima Homework Statement If a wave is reflected at a surface of a plate with higher refractive index (eg. At air glass or air-metal) it suffers a λ/2 phase change. Show that for the interferometer, maxima will occur for 2dcosθ = (n+1/2)λ and...
  23. Telemachus

    Hessian matrix, maxima and minima criteria

    Homework Statement Hi there. I've got some doubts about the maxima and minima on this function: f(x,y)=x \sin y. I've looked for critical points, and there's only one at (0,0). The thing is that when I've evaluate the second derivatives I've found that f_{xx}=0, then I have not a defined...
  24. Z

    Maxima and minima properties problem

    Homework Statement Let An be the nth term of an A.P. and if A7 = 15, then the value of the c.d. that would make A3 x A7 x A12 greatest is : 1)9 2)9/4 3)3/8 4)18 Homework Equations The Attempt at a Solution Applying AM>=GM A3+A7+A12/3 >= (A3 x A7 x A12)^1/3 given that...
  25. V

    Maxima and Minima application question

    Homework Statement A farmer is planning to build six adjoining rectangular pens of equal size to house his hens, as shown in the diagram. He only has 180m of fencing, however, and he wants to make the pens as large as possible. Find the maximum area he could make each pen. The diagram is...
  26. N

    Absolute vs. Relative Maxima and Minima

    Homework Statement I do not have a specific homework problem, but could someone please clarify this for me? QUESTION: When you have an absolute maxima (or minima), how can you tell if it is ALSO a relative maxima (or minima)? I understand how to find absolute extrema on a closed...
  27. Simfish

    Is the conjugate gradient algorithm susceptible to getting into local minima?

    What about the nonlinear forms of it? Or is it guaranteed to reach a global minimum?
  28. H

    Single slit diffraction minima formula misunderstanding

    Reading the derivation for the single slit diffraction minima formula from this site, I'm not convinced about its argument. http://www.math.ubc.ca/~cass/courses/m309-03a/m309-projects/krzak/index.html" What I'm having trouble understanding is the explanation of the formula for minima when m>1...
  29. H

    Maxima, Minima complex function

    Hey my problem is that I am unable to calculate the absolute value of the following function: f(z)=\bar{z}(z-2)-2\Re z wherase z=x+iy What i did was: =|z|^2-2\bar{z}-2\Re z=x^2+y^2-2x+2iy-2x=x^2+y^2+2yi-4x and how should i calculate the absoulte value of this function?? Because i should...
  30. D

    Maximum Slit width diffraction minima

    Homework Statement What is the maximum slit width so that no visible light exhibits a diffraction minimum?(Visible light has wavelengths from 400 nm to 750 nm.) Homework Equations D*sin(theta)=m*lambda ; where D = slit width,theta in degrees, m =1,2,3,..., and lambda is wavelength...
  31. T

    Maxima and Minima on surfaces in three dimensional space

    Homework Statement Find the maximum and minimum values of f(x,y) = (xy)2 on the domain x2 + y2 < 2. Be sure to indicate which is which Homework Equations I am not sure what to put here. I solved this problem a different way, and I am not confident I did it correctly. The Attempt at a...
  32. C

    Quadratic form (maxima and minima problem)

    Homework Statement Determine the largest value of x_{1}^2 + x_{2}^2 + x_{3}^2 when x_{1}^2 + 2x_{2}^2 + 2x_{3}^2 + 2x_{1}x_{2} + 2x_{1}x_{3} + 2x_{2}x_{3} = 1Homework Equations Not sufficiently relevant to produce the expected answer.The Attempt at a Solution Completing the square, we get...
  33. K

    Calculators Finding Maxima and Minima with TI-89

    I am new, but i have a question about the TI -89 Titanium calculator, does anyone how if i can plug in the equation f(x)=2x^3-15x^2+36x-1 to identify each location of the local maximun and minimum, and inflexion poits? Will this calculator do this?
  34. T

    Find Absolute Maxima and Minima of f(x) on [-2,2]

    1. Find the absolute maxima and the absolute minima of the following function f(x)=(2x)/(x^2+1) on [-2,2]
  35. U

    Calculating slit width by length difference of 1st and 2nd minima

    Homework Statement What is the slit width? Wavelength is 635 nm. Distance between screen and slit, 2.5 m. Distance on screen between first and second minima above the central maxima is delta 6 mm. Homework Equations D* sin theta * = m * lambda tan theta = x/ distance between screen...
  36. 3

    How Do You Determine Lines of Maxima and Minima Based on Gradient?

    Homework Statement Please look at the attached picture In that I had to draw the line of maxima and minima Homework Equations line of maxima has greatest gradient line of minima has smallest gradient The Attempt at a Solution which line should be the maxima? the line with the...
  37. S

    Maxima and Minima of z=f(x,y) theorems

    Hi all, In my Calculus III course, we are using Stewart's book, so as you might know there is not much rigor in there. Likewise, when it came to the section on Maximum and Minimum values of a function with two variables z=f(x,y), they ommited a lot of stuff. Hence, i tried to use...
  38. C

    Finding the Minimum Hypotenuse of a Right Angled Triangle | Hypotenuse Homework

    Homework Statement a and b are two perpendiculars drawn to the sides from a point on the hypotenuse of a right angled, isoceles triangle. Find the minimum value of the hypotenuse in terms of a and b. Homework Equations ABC is a right angled, isoceles triangle, with AC, the hypotenuse...
  39. E

    Optimization with maxima and minima

    A rectangle is to be inscribed in a right triangle having sides 6 inches, 8 inches, and 10 inches. Determine the dimensions of the rectangle with greatest area. I recently tried doing it and the answer was found by finding the slope and then using the first and second derivatives of the area...
  40. N

    Maxima and Minima of a function

    Are there any analytical techniques to do this besides the Derivative Test?
  41. A

    Confirming Stationary Minima & Convexity of a Function

    If the stationary points of a function are minimum points does that qualify the function to be a convex function? Also, if the function has only one stationary minimum point does that mean that that point is its global minimum? Can someone please confirm these for me Thank you
  42. Д

    How can I find the critical points of a function involving fractions?

    How will I find minima,maxima or both of one function (if they exist)? For example: f(x)=x^2-3x+2; f(x)=x^2-2x-x+2; f(x)=x(x-2)-(x-2); f(x)=(x-2)(x-1) x1,2=1,2 Those are the zeroes of the function. But how will I find the critical points (minima,maxima) of the function if x...
  43. B

    Diffraction Minima confusion-

    Homework Statement I did a physics lab the other day that dealt with two slit diffraction. In my setup, I also saw two faint diffraction minima, and it is my understanding that there is also some single slit diffraction effects. I measured the position a diffraction minimum to be y= 8.775...
  44. D

    Local Minima of y=x^3-2x^2-5x+2

    for the equation... y = x^3 - 2x^2 -5x +2 is its local minima at (2.120,-8.061) Thanks
  45. V

    A) Solving Interference Minima: Find Wire Diameter

    Homework Statement A)Two flat slides of glass are separated at one edge by a thin wire. The top surface of the upper slide and the bottom surface of the lower slide have special coatings on them so that they reflect no light. The system is illuminated with light of wavelength 418 nm. Looking...
  46. J

    Applied Maxima and Minima

    Homework Statement For a monopolist's product, the demand equation is: p = 42 - 4q and the average-cost function is c = 2 + (80/q) Find the profit-maximizing price. Homework Equations When i started to solve the problem, i deduced...
  47. C

    How Do Slit Separation and Width Affect Diffraction Patterns?

    Does the distance between the maxima increase/decrease or stay the same when the slit separation is increased? when the slit width is increased? what about if it was minima? thanks for any help to understand how this works.
  48. V

    Probability at Minima or Maxima

    I hate to make another topic so soon, but I'd like to get some opinions on this question... Homework Statement A two slit electron difraction experiment is done with slits of unequal widths. When only slit 1 is open, the number of electrons reaching the screen per second is 25 times the...
  49. I

    Charge a capacitor with minima dissipation

    I need to design a circuit to charge a capacitor with minima loss of dissipation through resistance. Any idea?
  50. J

    Applied Maxima and Minima Problem

    Homework Statement Problem goes like this: For a monopolist's product, the demand equation is: p=156-2q _ and the average-cost is c(ave)=120+112/q Homework Equations We need to find the total cost function in terms of c= The Attempt at a Solution...
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