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. Clemencio

    I Single-Slit Diffraction -- Deriving the relation for the minima

    Hi everyone. I'm studying single-slit diffraction, and a question came up: to derivate the relation for the minima (dark fringes), the slit is devided in two parts, and it's assumed the distance between the light rays is a/2. Why is this distance chosen? I was wondering about other options...
  2. chwala

    Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##

    The interest is on number ##4##, In my working, ##f(x,y,z) = x+y^2+2z## and ##g(x,y,z) = 4x^2+9y^2-36z^2 = 36## ##f_x = 1, f_y=2y## and ##f_z = 2## and also ## g_x = 8λx, g_y = 18λy## and ##g_z = -72λz## using ##\nabla f (x,y,z) = λ\nabla f (x,y,z)## i shall have, ##1 = 8λx ## ##2y =...
  3. chwala

    Find the local maxima and minima for##f(x,y) = x^3-xy-x+xy^3-y^4##

    Ok i have, ##f_x= 3x^2-y-1+y^3## ##f_y = -x+3xy^2-4y^3## ##f_{xx} = 6x## ##f_{yy} = 6xy - 12y^2## ##f_{xy} = -1+3y^2## looks like one needs software to solve this? I can see the solutions from wolframalpha: local maxima to two decimal places as; ##(x,y) = (-0.67, 0.43)## ...but i am...
  4. C

    Why is continuity necessary before applying the Extreme Value Theorem?

    For this problem, Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)? Many thanks!
  5. tworitdash

    A Finding Global Minima in Likelihood Functions

    I have a likelihood function that has one global minima, but a lot of local ones too. I attach a figure with the likelihood function in 2D (it has two parameters). I have added a 3D view and a surface view of the likelihood function. I know there are many global optimizers that can be used to...
  6. P

    I Exploring the Relationship Between Lattice Successive Minima and Basis Vectors

    Hello, I've been thinking a bit about the definition of the ##i##-th successive minima of a lattice (denoted with ##\lambda_i(\Lambda)##), and I would argue that the ##i##-th successive minimum is at most as large as the largest lattice basis vector ##b_i##. More formally...
  7. WMDhamnekar

    MHB Maxima and Minima (vector calculus)

    Hi, Hi, Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$ But when I read this graph, I couldn't guess that the stated function have a...
  8. WMDhamnekar

    MHB Maxima and Minima in calculus

    Question: Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone. Answer: Let r and h be the radius and height of the right circular cylinder inscribed in a given cone of radius R and height H. Let...
  9. A

    Finding Absolute Minima & Maxima of a Function

    Good day, I have a question regrading how to find the absolute minima maxima of a function , I understand that first we need to calculate the Hessian Matrix to find the relative minima /maxim but after we need to check the borders of the region ( a rectangle in our case) for example we put x=-2...
  10. Mayhem

    I Finding global minima of nth degree polynomials

    Is it possible (read: reasonably easy) to find global minima of an nth degree polynomial of the general form $$a_nx^n + a_{n-1}x^{n-1} ... a_2x^2 +a_1x + a_0 = 0$$ It seems to have applications in computational chemistry as I have a "hunch" that polynomial regression could be used to somewhat...
  11. M

    I Finding Multiple Local Minima of High-Dimensional Functions

    Hi all, Let me give some background to my question. In computational neuroscience it's now fashionable to train recurrent neural networks (RNNs) to solve some task (working memory for example). We do so by defining some cost function and training the model weights to minimize it (using...
  12. A

    Solve the Global Minima Problem in Two Variable Functions

    I'm always struggling understand how to determine if a two variable function has global minima, I know that if I find a local minima and the function is convex than the local minima is also a global minima, in this case is really difficult to determine if the function is convex. Sorry if I...
  13. T

    Image of a f with a local minima at all points is countable.

    Homework Statement Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
  14. Z

    Quotient Derivative and Minima Maxima

    Homework Statement Find out the quotient derivative i.e. the derivative of polynomial upon polynomial and then find the minima and maxima.[/B] ##W\left(z\right)=\frac{{4z+9}}{{2-z}}## Homework Equations ##\left( \frac{f}{g} \right)' = \frac{f'\,g - f\,g'}{g^2}## The Attempt at a Solution...
  15. T

    Finding the min value using the derivative

    Homework Statement Hi I'm having a trouble with finding min value of given function: f(x) = sqrt((1+x)/(1-x)) using derivative.First derivative has no solutions and it is < 0 for {-1 < x < 1} when f(x) is given for {-1 < x <= 1}. For x = - 1 there is a vertical asymptote and f(x) goes to +...
  16. jamalkoiyess

    I Delta x in the derivation of Lagrange equation

    Hello PF, I was doing the derivation of the Lagrange equation of motion and had to do some calculus of variations. The first step in the derivation is to multiply the integral of ƒ(y(x),y'(x);x)dx from x1 to x2 by δ. and then by the chain rule we proceed. But I cannot understand why we are...
  17. S

    Find all relative maxima/minima and saddle points

    Homework Statement f(x,y)=2x^3+xy^2+5x^2+y^2 Homework EquationsThe Attempt at a Solution fx(x,y)=6x^2+y^2+25x fy(x,y)=2xy+2y fxx(x,y)=12x+25 fyy(x,y)=2y+2 fxy(x,y)=2y I found the partial derivatives from the equation. I am stuck at finding the critical points from the first two fx(x,y) and...
  18. L

    I X-ray diffraction minima?

    For x-day diffraction maxima we have braggs law 2d*sinθ = mλ (maxima) Is there an analogous law for the minima like 2d*sinθ = (m+1/2)λ (minima?) Thanks!
  19. S

    Mathematica How do you find the number and location of local minima of a given function?

    Take the function $$f(x) = ax^{2} + bx^{4} - c \cos(x/d),$$ where ##a##, ##b##, ##c## and ##d## are arbitrary parameters. For some given choice of the parameters, how do you find the number of local minima of the function and the location of the minima?
  20. T

    MHB Maxima, minima, and the mvt application

    Hi there I'm prepping for a big test tomorrow and I'm really struggling with this question:If f′′(x)≥−1, x belongs to (−15,15), and f′(1)=3, find the interval over which x is definitely increasing.I'm struggling with substitution because I just don't seem to have enough values. Is there a...
  21. H

    I Derivative = 0 is always minima? (Linear variational method)

    I have a very fundamental question about the linear variational method (Huckel theory). It says in any textbook that the variational method provides energy upper bound to the actual energy of a wavefunction by using test wavefunction. \varepsilon = \frac{\sum_{i,j}^{n}C_{i}C_{j}H_{ij}...
  22. M

    MHB Are the critical points minima or maxima?

    Hey! :o We have the function $f(x_1, x_2)=2-x_1-x_2$ and we want to check if it has maxima or minima under the constraint $x_1^2+x_2^2=8$. Since we cannot solve for one variable at the equationof the constraint, we have to use the Langrange function, right? (Wondering) I have done the...
  23. Peter Andrews

    Finding the angular spread of a diffraction minima?

    1. Homework Statement Light of wavelength 6000Å illuminates a single slit of width 10-4m. Calculate the angular spread of first diffraction minima.Homework Equations d*y/D = nλ Y = nλ/a for minima Y = (2n±1)λ/a for maxima Y stands for the position on screen, d is slit width and D is separation...
  24. S

    A Spontaneous symmetry breaking - potential minima

    In spontaneous symmetry breaking, you expand the Lagrangian around one of the potential minima and write down the Feynman rules using this new Lagrangian. Will it make any difference to your Feynman rules if you expand the Lagrangian around different minima of the potential?
  25. nso09

    Three slit minima and maxima and path differences

    Homework Statement Find at which angles θ the interference picture that appears on a distant screen made by three thin slits separated by distance d and enlightened by a source of wavelength λ (see figure) a) Shows its maxima. b) Shows its minima. multiple slit diffraction, d, ##\lambda##...
  26. A

    Calculus of Variations: Functional is product of 2 integrals

    Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...
  27. S

    How do I solve systems of equations to find local max, min, and saddle points?

    Homework Statement Find the local max, min, and saddle point for the function: f(x,y) = 2x^2+3xy+4y^2-5x+2yHomework EquationsThe Attempt at a Solution I've taken the two partial derivatives Fx = 4x + 3y - 5 Fy = 3x + 8y + 2 I know that the critical points will sit where both of theses partial...
  28. Matejxx1

    Maxima and minima and finding the radius of the circle

    Homework Statement Find where θ is the biggest (largest) I'll have the picture of the problem included below (pic:1) Homework Equations (x-q)2+(y+5/2)2=r2 answer x= 2 The Attempt at a Solution Hi, so my prefesor gave me this problem and told me to try to solve it. We already did this problem...
  29. F

    Minima and Maxima of a multivariable function....

    Hello Forum, I think I am clear on how to find the maxima and minima of function of one or two independent variables like f(x) or f(x,y). What if the function had 5 independent variables, i.e. f(x,y,z,a,b)? What is the best method, even numerically? Should we find the partial derivatives of f...
  30. L

    MHB Maxima and minima of a triangle

    Write the following formulas: a) The minimum perimeter of any triangle (abc) only known heights corresponding to the sides a and b. b) The maximum height and minimum corresponding to the side b of any triangle (abc) only known the value of its perimeter and height corresponding to the side a.
  31. B

    MHB Finding the maxima and minima of a function

    I have the following function: f(x,y) = xye(-x2-y2) I am trying to find all the maximum and minimum points. I have started off by finding the partial derivatives of the function and equation them to zero. df/dx = ye(-x2-y2)(1-2x2) = 0 df/dy = xe(-x2-y2)(1-2y2) = 0 However, i am stuck for...
  32. S

    Dimensional analysis and minima of a potential

    Homework Statement Consider the Euclidean classical action ##S_{cl}[\phi] = \int d^{4}x (\frac{1}{2}(\partial_{\mu} \phi)^{2} + U(\phi))##, with ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##. (a) Show that, in four-dimensional space-time, the mass...
  33. L

    Finding the maxima or minima of band-edges?

    I'm working on a homework assignment and I'm completely stuck on this last problem. I'm not even sure what ∇KE=0 even means to begin with. I understand the difference between indirect and direct bandgap but I'm just confused on how to find the maxima and minima to determine this. Any help would...
  34. C

    Minima of a multivariable function.

    Homework Statement Find the minimum value of f(x,y)=e^{x+y}-2 within x≥0 and y≥0. Homework Equations D=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2 Answer is -1 The Attempt at a Solution So for all partial derivatives I got e^{x+y} (and mixed), but when I calculate the discriminate (subbing in...
  35. C

    Is this question missing a third time interval?

    Homework Statement Given critical points (3,-4) and (6,0); interval of increase (3, infinity); interval of decrease (-infinity, 3), find the local maxima/minima and sketch the graph. Homework Equations No relevant equations are given, I believe it's a simple sketch the graph. The Attempt at...
  36. A

    Minima of a diffraction grating

    Note from mentor: This thread was originally posted in a non-homework forum, so it does not use the homework template. -------------- We have n slits, however suppose half of the middle ones are covered. How could you go about finding the angles at which the minima occur at in the Fraunhofer...
  37. P

    Maxima and Minima, Rod Rotating on a Surface With Friction

    Homework Statement A uniform rod of mass M and length L can rotate around point P which is at position x from one end of the rod as shown in the figure. The rod is gently placed on a rough horizontal surface that has a friction coefficient μ. and at t=0 starts rotating with angular velocity...
  38. L

    Meaning of Single Photon Interference Minima

    In the double slit experiment, when we send out one photon at a time, what does the appearance of minima in our interference pattern mean? When a single photon is fired, I understand (using "understand" very loosely) that the photon will interfere with itself. When this happens, does it still...
  39. trash

    Maxima and minima of differential equation

    Homework Statement Consider the differential equation y'=x-y^2. Find maxima, minima and critical points; show that for every solution f=f(t) exists T\geq 0 such that f(t)< \sqrt{T}\;\forall t > T Homework Equations The Riccati equation: y'=a(x)y^2+b(x)y+c(x) The Bernoulli equation...
  40. D

    Path differenece of light ray to produce first minima

    Homework Statement for part a , in order to produce first minima from the central bright ray , the path differenece between 2 ray should be equal to λ/2 or 180 degree am i right? why the notes give the path differenece = λ ...If path differenece = λ , the two light ray are in phase right...
  41. M

    MHB Gibbs Phenomenon:The maxima get smaller,the minima get bigger

    Hey! :o I am looking at the Gibbs Phenomenon for the function $\displaystyle{f(x)=sgn(x), x \in [-\pi, \pi]}$. The Fourier series of this function is: $$sgn(x) \sim s(x)=\sum_{k=1}^{\infty} \frac{4}{(2k-1) \pi} \sin{((2k-1)x)}$$ Since $f$ is odd, it's sufficient to look its behaviour at $[0...
  42. C

    Proving the Local Maxima of a Rectangle Partition with Two Parallel Lines

    A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles. Unless I did incorrectly, the algebra is very very long... HELP
  43. S

    Identifying Critical Points in e^x(1-cosy): Local Extrema or Saddles?

    1. If f(x,y)=e^{x}(1-cos(y)) find critical points and classify them as local maxima, local minima, or saddle points. The Attempt at a Solution I found the partials and mixed partial for the second derivative test as follows: f_{x}=-e^{x}(cos(y)-1) f_{y}=e^{x}(sin(y))...
  44. J

    Find all critical points, and identify them as minima, maxima, or

    Homework Statement Consider the function f(x,y) = (x2 + 4y2)e(1-x2-y2) Find all critical points, and identify them as maxima, minima, or saddle points. The Attempt at a Solution I took the partial of x and the partial of y, and set them equal to 0. This is what I got: fx(x,y) =...
  45. S

    Maxima and Minima of a function of several variables

    Homework Statement Find the maxima and minima of: f(x,y)=(1/2)*x^2 + g(y) g∈⊂ (δ⊂ ℝ ) in this region Ω={(x,y)∈ℝ2 / (1/2)*x^2 + y^2 ≤ 1 } hint: g: δ⊆ ℝ→ℝ The absolute min of f in Ω is 0 The absolute max of f in Ω is 1 Homework Equations The Attempt at a Solution I have the...
  46. O

    Testing Algorithm for Global Minima of Test Functions

    I'm testing an algorithm to find the global mimina of a function. Can someone give me a few examples of optimization test functions in 2 or 3 dimensions, like the Rastrigin function. I'm hoping to find functions with several local minima.
  47. M

    Find global minimum of f(x1,x2, xn), with many local minima

    Hi I have some function f = f(x1,x2,...xn) over some domain [0,1]^n, and I'd like to find the global minimum. The function is *highly* non-linear and takes about 1 second to evaluate. I know it's positive because it's the sum of squares of about 1,000,000 arguments, each of which pretty much...
  48. C

    Maxima and Minima of two-variable functions word problem

    1. Problem: A manufacturer makes two models of an item, standard and deluxe. It costs $40 to manufacture the standard model, and $60 for the deluxe. A market research firm estimates that if the standard model is priced at x dollars, and the deluxe at y dollars, then the manufacturer will sell...
  49. L

    Optimizing Distance: Finding the Minimum Value and Coordinates on a Line

    A point P whose x-coordinates is a is taken on the line y=3x-7. If Q is the point(4,1) show that PQ2 = 10a2-56a+80. Find the value of a which will make the expression a minimum. Hence show that the coordinates of N, the foot of the perpendicular from Q to the line are (24/5 , 1 2/5). Find...