What is Extremum: Definition and 25 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. Reuben_Leib

    I Help with Euler Lagrange equations: neighboring curves of the extremum

    I tried writing this out but I think there is a bug or something as its not always displaying the latex, so sorry for the image. I have gone through various sources and it seems that the reason for u being small varies. Sometimes it is needed because of the taylor expansion, this time (below) is...
  2. 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 +...
  3. Math Amateur

    I Corollary to the Interior Extremum Theorem .... ....

    I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 6: Differentiation ... I need help in fully understanding the corollary to Theorem 6.2.1 ... Theorem 6.2.1 and its corollary ... ... read as follows: I am...
  4. Math Amateur

    MHB Understanding Corollary 6.2.2 of B&S Theorem 6.2.1

    I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 6: Differentiation ... I need help in fully understanding the corollary to Theorem 6.2.1 ...Theorem 6.2.1 and its corollary ... ... read as follows: Can someone...
  5. L

    B Understanding Extremum and Derivatives at Boundary Points

    It's understandable that finding absolute extremum is impossible for a function with restricted boundary conditions. But why does the derivative of similar functions is not zero when the extremum is on the end points? To be precisely short with my question, why does the derivative gives only...
  6. mr.tea

    I Lagrange multipliers and critical points

    Hi, I have (probably) a fundamental problem understanding something related critical points and Lagrange multipliers. As we know, if a function assumes an extreme value in an interior point of some open set, then the gradient of the function is 0. Now, when dealing with constraint...
  7. E

    I How to minimise this function?

    I tried calculating the partial derivative of ##\varphi\left(x, y\right) = \sum_\lambda\left\{H\left(\lambda\right) \left[C_E\left(\lambda; x, y\right) + \sum_n a_n\left(x, y\right) e_n\left(\lambda\right)\right]^2\right\}## with respect to ##a_n## and equating it to zero to minimise the...
  8. D

    Extremum of thermodynamic potentials: confusion

    An alternative formulation of the second law is that the energy of the system U is minimised if the temperature and entropy of the system are held constant. However, dU= TdS -pdV which means that U is presumably constant if the volume V and the entropy S are kept constant. How then can U...
  9. W

    Can curvature ever be greater than at relative extremum?

    Homework Statement For a generic function y=f(x) which is twice-differentiaable, is it possible for there to be a curvature on the curve of that function that is greater than the curvature at its relative extremum?Homework Equations The Attempt at a Solution From visualization and a sketch...
  10. Coffee_

    Proof: extremum has a semi definitie Hessian matrix

    Consider a function ##f : U \subseteq \mathbb{R}^{n} -> \mathbb{R}## that is an element of ##C^{2}## which has an minimum in ##p \in U##. According to Taylor's theorem for multiple variable functions, for each ##h \in U## there exists a ##t \in ]0,1[## such that : ##f(p+h)-f(p) =...
  11. nuuskur

    Proof: Local extremum implies partial derivatives = 0

    Homework Statement Let f\colon\mathbb{R}^m\to\mathbb{R}. All partial derivatives of f are defined at point P_0\colon = (x_1, x_2, ... , x_m). If f has local extremum at P_0 prove that \frac{\partial f}{\partial x_j} (P_0) = 0, j\in \{1, 2, ..., m\} Homework Equations Fermat's theorem: Let...
  12. T

    Finding the functional extremum

    Homework Statement I have been given a functional $$S[x(t)]= \int_0^T \Big[ \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)\Big] dt$$ I need a curve satisfying x(o)=0 and x(T)=1, which makes S[x(t)] an extremum Homework Equations Now I know about action being $$S[x(t)]= \int_t^{t'} L(\dot x, x)...
  13. A

    MHB Extremely Difficult Extremum Problem.

    Here is goes, Find the minimum value of $y = \left| sin(x) + cos(x) + tan(x) + sec(x) + csc(x) + cot(x) \right|$ I don't even know where to begin.
  14. D

    Show extremum of the Entropy is a maximum

    Homework Statement The Entropy of a probability distribution is given by, S = -k_B \sum _{i=1}^N p(i)\ln{p(i)} I've shown that the extremum of such a function is given by, S' = k_B \ln{N} (which is a positive quantity) Now I want to show that this is a maximum by showing that S' - S...
  15. Saitama

    How many extremum does the function have for given values of a and b?

    Homework Statement For all ##a,b \, \in \, R##, the function ##f(x)=3x^4-4x^3+6x^2+ax+b## has: a) no extremum b) exactly one extremum c) exactly two extremum d) three extremum Homework Equations The Attempt at a Solution ##f'(x)=12x^3-12x^2+12x+a=12x(x^2-x+1)+a## If a=0...
  16. R

    Simple Thermodynamic extremum problem

    Suppose I have a cylinder with a movable partition inside, separating it into two subsystems. The partition is movable, but will not transmit heat or matter. The same type of gas is contained in each subsystem, but the pressures and temperatures are different. The same amount of mass (M) is in...
  17. S

    Obtain the extremum of f(x,y,z)

    Homework Statement Obtain the extremum of f(x,y,z) = 2x^2 + y^2 + 2z^2 + 2xy + 2xz + 2y + x - 3z - 5 and determine its nature. Homework Equations Partial differentiation and systems of equations. The Attempt at a Solution My attempt is attached. In addition to confirming if what I...
  18. H

    Mean Curvature at Extremum Points on a Closed Surface

    Hi, I know that the mean curvature at an extremum point where the function vanishes must be nonpositive.can this say something about the sign of the mean curvature at the farthest point on a close surface from the origin? Thank's Hedi
  19. S

    Extremum Function of a Functional

    Homework Statement J(f)=\int 2xf−f′2+3f2f′dx f(0)=0,f(1)=−1. Homework Equations Ff-\frac{d}{dx}Ff'=0 The Attempt at a Solution Ff=2x+6f f'' Ff'=-2f' + 6f2 Plugging in, I get: 2x+6f f''- [itex]\frac{d}{dx} (-2f' + 6f2) 2x+6f f''-12f f'-2f''=0 Which doesn't look...
  20. Z

    Finding Extremum Points of f(x): A Study

    Homework Statement Let f(x) =(x-1)p.(x-2)q where p,q>1. Each critical point of f(x) is a point of extremum when - (Options are given) The Attempt at a Solution I got the critical points as 1 and 2. I don't know what do I do next. I found the second derivative but I think its of no...
  21. M

    Finding Extremum of z=1-sqrt(x^2+y^2)

    Homework Statement Find exterma points of: z=1-\sqrt{x^2+y^2} Homework Equations Second derivative test. The Attempt at a Solution I find that (0,0,1) is a point where an extremum exist. To determine whether it's a maximum or minimum I need to use the second derivative test, but my second...
  22. O

    Are local extremum possible at endpoints of a closed bounded interval?

    I thought local extremum did not exist at the endpoints of a closed bounded interval, however my textbook claims this. Wikipedia: "A continuous (real-valued) function on a compact set always takes maximum and minimum values on that set. An important example is a function whose domain is a...
  23. S

    Boundary considerations in extremum problems

    Homework Statement We are given a word problem and asked find maxima/minima (ie a simple example would be to find the least amount surface area required to build a box of a given volume). Is it necessary to explicitly show that the relative interior max/min, calculated by setting the gradient...
  24. M

    Local Extremum of f(x)=(4-x^2)^(-1/2): First Derivative Test

    1. According to the First Derivative Test for local extrema, if f' doesn't change sign at c, then f has no local extreme value at c. But for a question on my book, f(x)=(4-x^2)^(-1/2), the critical point is 0, but i think it doesn't have local extreme because the derivative doesn't change sign...
  25. E

    Is There an Alternative Method to Obtain the Extremum of a Functional?

    Let,s suppose we have a functional J and we want to obtain its extremum to obtain certain Physical or Math properties: \delta{J[f(x)]}=0 Yes you will say to me " You can apply Euler-Lagrange Equation to it and generate a Diferential equation to obtain f"..of course is easier saying than...