Questions tagged [maxima-minima]

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 of a function (the global or absolute extrema).

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Min-max principle for eigenvalues in 1d elliptic problem

I have the following eigenvalue problem: \begin{equation} \begin{cases} -u''=\lambda u\\ u(0)=u(\pi)=0 \end{cases} \end{equation} and I have to prove the following min-max priciple for eigenvalues: \...
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Show that the absolute maximum of $f(x, y) = \frac{(ax+by+c)^2}{x^2+y^2+1}$ is $a^2 + b^2 + c^2$

I got this question on my exam today: show that the function $f(x, y) = \dfrac{(ax+by+c)^2}{x^2+y^2+1}$ has an absolute maximum whose value is $a^2 + b^2 + c^2$. I tried setting the gradient to the ...
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Prove that $\Delta u=F$ with these conditions has at most one solution

Let $\alpha >0$, and let $\Omega\subset \mathbb{R}^N$ be and open domain. I want to prove that the following problem has at most one solution. $$\Delta u=F \quad \text{in } \Omega$$ $$u=f \quad \...
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Second partial derivative: What happens if $f_{xy}^2$ is larger than $f_{xx} f_{yy}$

Considering $(a,b)$ is a critical point of a funciton $f(x,y)$ and $D(x,y) = det(H(f(x,y))) =f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2}$ is the determinant of the hessian matrix from that function. If ...
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Knowing that $i\frac{z-z_4}{z_2-z_3},i\frac{z-z_5}{z_3-z_1},i\frac{z-z_6}{z_1-z_2}\in \mathbb R$, determine $\min(|z-z_4|^2+|z-z_5|^2+|z-z_6|^2)$.

Consider complex numbers $z_1 = 1 + i, z_2 = 1 - 3i, z_3 = 4 + i$ and complex variable $z$. Knowing that there exist complex numbers $z_4, z_5, z_6$ such that $\dfrac{z_4 - z_2}{z_4 - z_3}, \dfrac{z_5 ...
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Is a planar square on the equator a locally energy minimizing configuration of electrons on $\mathbb{S}^2$?

$\newcommand{\S}{\mathbb{S}^2}$Let$$M=\{(x_1,x_2,x_3,x_4) \in \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ Let $...
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Maximizing $P(t):=\frac{A^{2}}{N} \sum_{j=1}^{N} \sum_{k=1}^{N} x_{k} x_{j} \exp \left(2 \pi i \frac{B}{N}(k-j) t\right)$

$$P(t):=\frac{A^{2}}{N} \sum_{j=1}^{N} \sum_{k=1}^{N} x_{k} x_{j} \exp \left(2 \pi i \frac{B}{N}(k-j) t\right)$$ Could any one tell me how to maximize $P(t)$? $t\in [0,T)$. I have done the ...
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Maximize $(z-x)$ such that $x^2 + y^2+ z^2 =1$

Here I tried the coordinate geometry, as in the equation represents a sphere. From there $(z-x)$ would imply the distance between the $z$ coordinate and $x$ coordinate so that the difference is ...
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Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?

I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals) $$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\ g(x)=\frac{\...
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Find the minimum of $f(x)=x^2-x+1+\sqrt{2x^4-18x^2+12x+68}$.

WA gives the result $9$. But how to solve it by applying inequalites?
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Theorem of the Maximum for discrete sequences of constraint sets?

Suppose that $\{X_{n}\}_{n=1}^{\infty}$ is a sequence of sets that converges to $X$ in some sense. Let $f$ be a real-valued function. I am interested in conditions under which $$ \lim_{n \rightarrow \...
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Notation: minimum for all (sub) elements

Have an element list \Delta, these Elements have sub values, \alpha, \beta, ...
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Is the minimized sum greater than the sum of the minimum values?

Does $$\min_{x \in X} \sum_{t=1}^n f_t(x) \geq \sum_{t=1}^n\min_{x \in X} {f_t(x)}$$ hold? This is a conclusion I want to use and I feel it's right but I can't prove. Is it true? Note: f is convex in ...
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find the minimum product of two numbers whose difference is 17. What are those two numbers. [closed]

minimum product of two numbers whose difference is 17. What two numbers and how do I get it? what formula? .
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Maximum rectangle within a parallelogram

There is a quadrilateral with equal-length for opposite sides but the diagonals are different (and I hope the word parallelogram is correct here), what would be the biggest rectangle I can inscribe, ...
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necessary condition of strict local minimizer

Given a function $f(x)$ which has up to 2 order continuous derivatives. It is known that if $f(x)$ attains local minimum at $x=x_0$, then $f'(x_0)=0$ and $f''(x_0)\geq 0$. I am wondering what is the ...
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Recommendations to minimize unknown function with categorical variables

Goal The goal is to minimize an output variable of a computer code whose function is unknown, we will call it F. There are 25 input variables, xij, and 5 output variables, yk. xij, where i ∈ [1,5] ...
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Is there a general term that can be used for critical points not maximum or minimum?

Most references do not use or avoid using terms when discussing critical points that are not maximum or minimum. But for the sake of asking a general term that encompasses critical points which are ...
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Arrange N unit squares in form of a grid such that number of rectangle is maximum?

You are given N square tiles of dimension 1×1. You have to arrange them in form of a grid such that total number of rectangle (of all possible dimensions) is maximum. Hollows within the grid are not ...
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Compute sup norm of sequence of functions.

Consider the operator $T: C^2[0,1] \subset C^1[0,1] \to C^1[0,1]$ defined by $Tf=f'+f''$. Compute $\| T e^{-nx } \|_{\infty }$ and $\| T x^n \|_{\infty }$. My attempt. First I tried to compute $\| e^{...
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Alternate approach for finding Minimum value of the below function

If $M (x)$ = max ($4-x$, ($\frac{\sqrt x^{3}}{\sqrt3^3}$) ) , $0< x \leq 4$ find minimum value of $M(x)$ in $(0 ,4]$. what i did was first of all we need to find the x possible values for which $...
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what is the difference between **isolated local minimum** and **local minimum and isolated critical point**?

In section four of the paper On the stable equilibrium points of gradient systems, authors introduce two concepts listed below. I am confused about the difference between them. Could anyone help to ...
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Maximum of $\frac{Bx}{x^3 + Rx + P}$

I submitted this function to Wolfram Alpha and similar tools, but none can find for me the symbolic expression for maximum value, unless I give specific values to each parameter: $$\frac{Bx}{x^3 + Rx +...
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Minimum of $\sum_{i=1}^n \frac{1}{\prod_{j\ne i} |x_j - x_i|}$ for $x_1,...,x_n \in \mathbb [-1, 1]$

Let's consider $x_1,\cdots,x_n \in \mathbb [-1, 1]$. I want to find minimum value of: $$\frac{1}{|x_1-x_2|\cdots|x_1 - x_{n}|} + \frac{1}{|x_2-x_1||x_2-x_3|\cdots|x_2 - x_n|}+\cdots+\frac{1}{|x_n-x_1|\...
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Find the minimum of a function containing the norm of another function

This is my first post here, I've been struggling with a problem for maths course, and would appreciate any guidelines on how to start tackling it (not asking for the answer, just some guidelines on ...
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Finding the supremum, infimum, and bounds of $f(x) = x^2$ for $x \le a$ and $f(x) = a + 2$ for $x > a$ in the interval $(-a-1,a+1)$, where $a > -1$.

My current progress; $x^2$ is increasing, so it will always have a minimum of $0$. Furthermore, if $a > -1$, then $a + 2 > 1$ and $-a -1 < 0$. Therefore, $f(x) \ge 0$. I tried to split $f$ ...
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When will the minimiser of a convex objective not be unique?

Given a response $Y\in\mathbb{R}^n$ and design matrix $X\in\mathbb{R}^{n\times p}$ consider the regression estimator $$\hat{\beta}=\text{arg min}_{\beta\in\mathbb{R}^p}\frac{1}{2n}\|Y-X\beta\|_2^2+\...
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Knowing that $f(x) = ax^4 + bx^3 + cx^2 + dx + e \ (a \ne 0)$ and $e > n$, how many extrema does the function $y = f'(f(x) - 2x)$ have?

Consider graph $f(x) = ax^4 + bx^3 + cx^2 + dx + e \ (a \ne 0)$ whose derivative's graph is illustrated as the following. Knowing that $e > n$, how many extrema does the function $y = f'(f(x) - 2x)$...
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Find the parameters $a,b$ such that the distance of all the points of a subset of $\mathbb{R^2}$ to the line $y=ax+b$ is minimal

Let $A\subset \mathbb{R^2} $ be a non-empty, finite set. We define a function $f:\mathbb{R^2}\rightarrow\mathbb{R},$ $f(a,b)=\sum_{(x,y)\in A}||y-ax-b||^2$. Find the global minimum of this function. ...
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Determine the minimum/maximum value of expression $P = MA + 2NB + 4MN$ where $A=(\frac{17}{4}; 0; 0), B=(5; 4; 0)$ and $M$ and $N$ are mobile points.

Consider points $A\left(\dfrac{17}{4}; 0; 0\right), B(5; 4; 0)$ and spheres $(C_1)\colon \, x^2 + y^2 + z^2 = 1, (C_2)\colon \, x^2 + (y - 4)^2 + z^2 = 4$. Given points $M$ and $N$ mobile on spheres $(...
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Calculate the value of expression $Q = \frac{x + 1}{y}$ when $xy > 1$ and expression $P = x + 2y + \frac{5x + 5y}{xy - 1}$ reaches its maximum value.

Consider two positives $x$ and $y$ where $xy > 1$. The maximum value of the expression $P = x + 2y + \dfrac{5x + 5y}{xy - 1}$ is achieved when $x = x_0$ and $y = y_0$. Calculate the value of ...
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Find the phase shifts that minimize the amplitude of the sum of a few sine waves of different period and amplitude?

I need to find the phase shifts that minimize the amplitude of the sum of a few sine functions. Example: $y_1 = max(a_1 * sin(b_1x + c_1))$ $y_2 = max(a_2 * sin(b_2x + c_2))$ ... $y_n = max(a_n * sin(...
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Argmax of the product of positive functions [closed]

Let $f(x), g(x) \geq 0$. Then, I want to know if the following is true $$ \arg \max_x [f(x)g(x)] = \arg\max_x f(x) \cdot \arg\max_x g(x) $$ And how one can prove it. I found a related question in this ...
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Determine minimum values of a function

Find the $x$ values that minimize the function $f(x)$. $$ f(x) = e^x + \frac{1}{m}(x-4)^2 $$ I know I can determine the minimum values when the derivative of the function is $0$. So I have calculated ...
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Continuity of a pointwise maximum function of probability distributions

Consider a pair of finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. Let $P_{Y|X}$ be a conditional probability distribution and let $Q_Y$ be a full-rank probability distribution. I am looking at the ...
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Minimize a function regarding two coupling variables

Given known matrics $A\in \mathbb R^{2\times 2}$ and known vectors $b\in \mathbb R^2, c\in \mathbb R^2$, for the two optimization variables $x\in \mathbb R^2$ and $y\in \mathbb R$, how to obtain the ...
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Claim concerning the maximum and minimum of $ \sin x \,\cos A+ \sin y\, \cos B$

I have this expression $$ \sin x \,\cos A+ \sin y\, \cos B \, ,$$ with $x,y\in\mathbb{R}$ and $0\leq A,B\leq2 \pi$. Then based on $ ∣\cos A∣\leq 1$ and $ ∣\cos B∣\leq 1$, is this claim true for the ...
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Optimization doubt

Let $f(x)=x^2+ax+b$ is a function $a,b \in \mathbb R, a^2>4b$ and $g(x)=x^2+2x-1 $ such that $f(g(x_i))=0\ \forall\ i\in \{ 1,2,3,4\}$ and $x_i<x_i+1\ \forall \ i\in \{1,2,3 \}$ and $x_1,x_2,...
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Bound on amount of minima of finite Fourier sum

Consider a finite Fourier sum of the form $$f(\theta) = \sum_{i=1}^n r_i \cos(m_i \theta) \,,$$ where $n \geq 1$ is an integer, the $r_i$ are positive real numbers and the $m_i$ are integers. Is there ...
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find maximum of product of two numbers.

Let $n=100, r=4$. Let $d=[2,7, 17, 22, 47]$. For each $d$, let $l= \frac{n}{r+d-1}$ and $t=\lfloor(\frac{r+d-2}{r})\rfloor$. Here as the value of $d$ increases $l$ decreases and $t$ increases. And the ...
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3 votes
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Best way to remove a local maxima from a piecewise linear function

Let $x_1,...x_K \geq 0$ and $f$ be the piecewise-linear function given by $f(k)=x_k$ for every $1 \leq k \leq K$. Denote by $m$ the number of modes (i.e. local maxima) of $f$. Let's associate with ...
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Shortest distance between two joggers

Question Two joggers, $A$ and $B$, start at either end of a $100$ m track. $A$ runs horizontally towards the other end at $3$ m/s and $B$ runs diagonally at $5$ m/s. The diagonal forms an angle of $30$...
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2 votes
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Why is minimax always greater or equal to maximin?

Let $f(x,y)$ be a function representing a game, we define the maximin : $\underline{f} = \max _{x∈X} \min _{y∈Y} f(x, y)$ and the minimax $\overline{f} = \min _{y∈Y} \max _{x∈X} f(x, y)$. It is said ...
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Of a minima problem in optics

I have trodding through a calculus textbook, more specifically — through a chapter on the methods of obtaining the extrema of functions using derivatives, including certain problems in optics (Fermat’...
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What is max[$XY + YZ + ZX$] if $X^2 + Y^2 + Z^2 =1$?

For real $X,Y,Z$, how to find the maximum of $XY+YZ+ZX$ subjected to condition $X^2 + Y^2 + Z^2 =1$? I am aware of the fact that for a single variable function $f(x)$, one would simply find the ...
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Of a minima problem concerning the Principle of the Parabolic Mirror

I have been trodding through the Course in Differential and Integral Calculus offered by R. Courant (Vol. I, Second Edition, 1954), specifically through the section on determining the maxima/minima of ...
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2 votes
1 answer
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Finding the minimum value of $\frac{p}{q-r}$, where $(-1,p)$, $(0,q)$, $(1,r)$ lie on parabola $y=ax^2+bx+c$, with certain conditions

If the vertex of the parabola $$y=ax^2+bx+c \qquad (0<2a<-b)$$is not below the x-axis. Let there be three points on the parabola: $A(-1,p), B(0,q)$ and $C(1,r)$. Then what is the minimum value ...
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2 votes
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Does the electrostatic potential have a local maximum on the sphere?

Let $$M=\{(x_1,x_2,x_3,x_4) \in \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \times \mathbb{S}^2 \, |\,\, \text{ all the } x_i \, \text{ are distinct}\} $$ $M$ is an open subset of $( \mathbb{...
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extremum points of function

I was wondering if I have a function F[x,t], (in my case a polynomial), and I find the extremum points, which are fractions, if the denominator of the extremum points vanishes, does this ensure ...
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Maximize $\sum_k \frac{p_k}{\sum_{j \geq k} p_j}$ over the probability simplex?

Suppose that $p_1, \dots, p_n$ are nonnegative real numbers such that $p_1 + \cdots + p_n = 1$; denote the corresponding set of vectors by $\Delta_n$. I am interested in the following function, $f \...
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