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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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If $K\subset \mathbb{R}^2$ is closed, can we find a smooth function $f:U\to \mathbb{R}$, s.t. $U$ is open, $K\subset U$ and $f$ is "nowhere" constant?

In John Lee's Introduction to Smooth Manifolds, p. 47, we have the following result: Theorem 2.29 (Level Sets of Smooth Functions). Let $M$ be a smooth manifold. If $K$ is any closed subset of $M$, ...
Derso's user avatar
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Is it correct to follow argmax with an inequality? $\arg \max_{x}\{f(x)<g\}$

In some papers, I saw a formula like this:$\arg \max_{x}\{f(x)<10\}$. I guess the author here wants to express that when $f(x)<10$, the largest of all $x$. This is different from the definition ...
贺星宇's user avatar
1 vote
2 answers
100 views

Prove maximum of $p\cdot(1-p)^{n-1}$ is at $p=\frac{1}{n}$ without differentiation

I would like to do a proof with my high school students that for $0\le p\le 1$, the maximum of $p\cdot(1-p)^{n-1}$ is reached for $p=\frac{1}{n}$. Proving this with differentiation is quite trivial, ...
Daniele's user avatar
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1 answer
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Find max value of $P = \frac{a}{2a + b ^ 2 + c ^ 2} + \frac{b}{2b+c^2+a^2} + \frac{c}{2c + a ^ 2 + b ^ 2}$

For $a,b,c \in \mathbb{R^+}$, $a+b+c=3$. Find max value of $P = \frac{a}{2a + b ^ 2 + c ^ 2} + \frac{b}{2b + c ^ 2 + a ^ 2} + \frac{c}{2c + a ^ 2 + b ^ 2}$ Have: $(2a+b^2+c^2)(\frac{1}{a}+a+1+1)\ge(1+...
trum fi fai's user avatar
4 votes
4 answers
288 views

Maximum of coefficients of a quadratic equation

Let $f(x)=ax^2+bx+c$ ($a\ne 0$) be a quadratic function. Given a specific $n\in\mathbb N_+$, $f(x)$ satisfies that for all $x\in[-n,n]$, $|f(x)|\leq n$. Find the maximum of $|a|+|b|+|c|$. My initial ...
Flaming's user avatar
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Find the minimum value of the sum of the squares of the distances from M to the lines AB, AC and BC

The problem Let ABCA'B'C' be a regular triangular prism with base edge $AB = 2 \sqrt{3}$ cm and height AA' = 1 cm. If M is a point in the plane of the triangle A'B'C' 0 , then the minimum value of the ...
Pam Munoz Ryan's user avatar
2 votes
2 answers
91 views

If $x > 0$ is a real number and $x+\frac{1}{x} \leq 7$ then the maximum value of the expression $x^2-\frac{1}{x^2}$ is...?

The problem If $x > 0$ is a real number and $x+\frac{1}{x} \leq 7$ then the maximum value of the expression $x^2-\frac{1}{x^2}$ is...? my idea We know that $x^2-\frac{1}{x^2}= (x- \frac{1}{x})(x+ \...
IONELA BUCIU's user avatar
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solution-verification | What is the maximum number of distinct planes determined by three of the 25 points?

The problem Consider a plane $\alpha$, a line $d || \alpha$, five points $A,B,C,D,E$, any 3 non-collinear located on the $\alpha$ plane and the points $P_1,P_2,... , P_20$ distinct two by two, located ...
IONELA BUCIU's user avatar
-1 votes
2 answers
85 views

Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$

The problem Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$ My idea $(\sqrt{a}+\sqrt{b})^2=a+b+ 2\cdot \sqrt{ab}= 101 + 2\cdot \sqrt{ab}$ ...
IONELA BUCIU's user avatar
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2 answers
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better method of solving quadratic / cubic

Problem : Let $$\begin{align} f(x) &= x^4 - 8x^3 + 18x^2 \\ g(x) &= 9x^2 - 64x\end{align}$$ . Define $h : \mathbb{R}^+ \to \mathbb{R}$, $h(x)=f(x)-ag(x)$ for some real number $a$. If $h(x)$ ...
bFur4list's user avatar
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A trigonometric maximum problem involving trigonometric constraints

Let $a,b,c,\alpha,\beta\in\mathbb{R}^+$ and $\alpha+\beta<2\pi$. Prove that if and only if $$\frac{\sin\alpha}{a\sqrt{b^2+c^2-2bc\cos\alpha}}=\frac {\sin\beta}{b\sqrt{a^2+c^2-2ac\cos\beta}}=-\frac{\...
Mr.He's user avatar
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3 answers
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Finding the maximum of $f(x) = \lim_{n \to\infty}n(x^{\sqrt[n]{x}}-x)$

What is the maximum of the function $$f(x) = \lim_{n \to\infty}n(x^{\sqrt[n]{x}}-x)$$ for $0 \leq x \leq 1?$ I know that $f(e) = e$ and $f(1/e) = 1/e$ (where $e$ is Euler's constant). The maximum is ...
codebender's user avatar
0 votes
2 answers
75 views

Show that $S(M) \leq 5a^2$

The problem Let $ABCDA'B'C'D'$ be a cube of edge a. On $[BC']$ we consider a point $M$ and write $S(M)=AM^2+CM^2+D'M^2$ a) Show that $S(M) \leq 5a^2$ b) Determine the position of point $M$ so that $S(...
IONELA BUCIU's user avatar
1 vote
0 answers
72 views

Prove that $\displaystyle \sum\limits_{i=1}^3\sqrt{ \sum\limits_{j=1}^3a_{ji}^2}\leq\sqrt{2}f(a_{11},\cdots,a_{33})$

For a $3\times3$ matrix $A=(a_{ij})$, let \begin{aligned}&f(a_{11},a_{21},a_{31},a_{12},a_{22},a_{32},a_{13},a_{23},a_{33})\\=&\text{max}\{|a_{11}+a_{21}+a_{31}|+|a_{12}+a_{22}+a_{32}|+|a_{13}+...
grj040803's user avatar
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2 answers
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Find min and max of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$

For $a,b,c \in \mathbb{R}$, $a^2 + b^2 + c^2 ⩽ 2$ Find min and max value of $P = (|a − b| + 3)(|b − c| + 3)(|c − a| + 3)$ I don't understand how to find min and max value of an absolute value sign. ...
trum fi fai's user avatar
1 vote
1 answer
48 views

solution-verification | Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$

the problem a) Show that, for any real number $x$, $x^4-4x^3+4x^2+3>0$ b) Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$, where x is a number real my idea So ...
IONELA BUCIU's user avatar
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1 answer
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Linearized formulas and constraints [closed]

I am researching how to linearize the maximum and minimum functions in optimization problems and have encountered some confusion. I am linearizing an equation, and ...
user1353491's user avatar
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0 answers
43 views

How to resolve symbolically "Maximum value of $2$ variable function $f(u,v)=\frac{\left(1-\sqrt{uv}\right)^2}{\frac{1-u^2}{2u}+\frac{1-v^2}{2v}}$"

In my answerto "Maximum value of $2$ variable function $f(u,v)=\frac{\left(1-\sqrt{uv}\right)^2}{\frac{1-u^2}{2u}+\frac{1-v^2}{2v}}$", I arrive at a numerical solution that the maximum is $f(...
Stephen Elliott's user avatar
3 votes
0 answers
83 views

When is $\max_a \max_b =\max_b \max_a $ [duplicate]

Lets consider a real valued and continuous function $f$ which takes 2 objects as input. If $f$ is symmetric: $f(a,b) = f(b,a)$ then I can obviously say: $$\max\limits_{a\,\in A} \max\limits_{b\, \in B}...
v.tralala's user avatar
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Calculate optimal spacing for magnetic field measurement using Gaussian Multivariate likelihood distribution

I posted this question on Physics exchange as well, but is rather mathematic :) I have a vertical magnetometer configuration, and measure lines on the ground. I want to calculate the optimal spacing, ...
user387449's user avatar
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0 answers
33 views

Minimization of a function with exponential and 2nd degree polynomial

I would like to minimize the following function (from $\mathbb{R}$ to $\mathbb{R}$): $$f(x) = -2a(1+x)\exp(-\Vert xb-c \Vert^2) + (1+x)^2 d$$ where $a,d \in \mathbb{R}$ and $b,c \in \mathbb{R}^d$ So ...
Cantor's user avatar
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3 votes
1 answer
145 views

Maximum value of $\displaystyle f(x)=\bigg|\sqrt{\sin^2(x)+2a^2}-\sqrt{2a^2-3-\cos^2(x)}\bigg|$

Maximum value of $\displaystyle f(x)=\bigg|\sqrt{\sin^2(x)+2a^2}-\sqrt{2a^2-3-\cos^2(x)}\bigg|$ What I try : Put $\displaystyle 2a^2-\cos^2(x)=t$ So we have $\displaystyle f(t)=\sqrt{1+t}-\sqrt{t-3}$...
jacky's user avatar
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0 votes
0 answers
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Maximum of two semimartingales

I get curious after reading this post on maximum of two Brownian motion: Process properties of the maximum of two independent linear Brownian motions. It occurs that for two stochastic processes $A=(...
John He's user avatar
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6 votes
3 answers
369 views

What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?

We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$ is $n$ (thanks to this question), but what is the global maxima of $\frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}...
RajaKrishnappa's user avatar
2 votes
6 answers
231 views

Maximum value of $2$ variable function $f(u,v)=\frac{\left(1-\sqrt{uv}\right)^2}{\frac{1-u^2}{2u}+\frac{1-v^2}{2v}}$

Finding maximum value of $\displaystyle f(a,b)=\frac{\bigg(1-\sqrt{\tan\frac{a}{2}\tan\frac{b}{2}}\bigg)^2}{\cot a+\cot b}$, Where $a,b\in\bigg(0,\frac{\pi}{2}\bigg)$ What I try : $\displaystyle \...
jacky's user avatar
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0 votes
1 answer
28 views

solution-verification | find a and then compare it to b

the problem a) Find the real numbers $x,y$ for which the number $a$ has the minimum value, where $a=\sqrt{4x^2+y^2-12x-4y+25}$ b)compare number $b=\sqrt{13+4\sqrt{3}}$ with the minimum value of $a$ my ...
IONELA BUCIU's user avatar
1 vote
1 answer
82 views

How to maximize $\mathbf{E}(XY) $ if we know the distribution of $X$ and $Y$

Given two discrete random variables (X) and (Y) with known distributions. The contingency table is provided: \begin{array}{c|ccc|c} & Y=1 & Y=2 & Y=3 & \text{Total} \\ \hline X=1 &...
Noname's user avatar
  • 585
17 votes
5 answers
564 views

Finding $a$ that minimizes the maximum value of $f(x)=\cos(2x)+\cos(x+a)$

For any real number $a$, let $f(x)=\cos(2x)+\cos(x+a)$ be a function with respect to $x$. Find $$\min\limits_{a\in\mathbb{R}}\max\limits_{x\in\mathbb{R}}f(x)$$ Or, in plain words, find the value of $a$...
Boopy's user avatar
  • 181
0 votes
1 answer
43 views

This function has no saddle points: correctness of this reasoning

I would like to know if my reasoning is correct. I have the function $f(x, y) = e^{3x}(1+25x^2+25y^2)$ and I have to study the stationary points. After computing the gradient I found $$\begin{cases} 3 ...
Heidegger's user avatar
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0 votes
0 answers
36 views

Necessary condition for unique point of minimum

My objective the following: Consider a function of two variables $f_{\xi}(x,y):\mathbb{R}\times\mathbb{R}^+\rightarrow\mathbb{R}$ which depends on a parameter $\xi \in \mathbb{R}^+$, where $f_{\xi}(x,...
stirling's user avatar
2 votes
0 answers
33 views

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius

What is the maximum area of n non-overlapping equal area triangles inscribed in a circle of radius 1? For n = 1, the triangle is equilateral. For n = 2, we have 2 isosceles right triangles sharing a ...
Ultima Gaina's user avatar
2 votes
5 answers
217 views

Find maximum value of $f(x,y)$ where $f(x,y)=\sqrt{x^2+2x+a}+\sqrt{y^2+2y+b}$

Given that $x\geq 0,y\geq 0,x+y=10$ and $a+b=132$. Let $$f(x,y)=\sqrt{x^2+2x+a}+\sqrt{y^2+2y+b} \tag{1}$$ If minimum value of $f(x,y)$ is $20$ then find the maximum value of $f(x,y)$. My Attempt $f(x,...
Maverick's user avatar
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0 votes
0 answers
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What is the intepretation of a Hessian with only negative and zero eigenvalues?

I'm looking at a high-dimensional optimisation problem. Specifically, one involving a function $F(c_n)$ which depends on N parameters (in my case $N=200$). I have been able to minimise this function ...
anonymous2506's user avatar
0 votes
1 answer
24 views

Proving an inequality with one constraint given

If $a,b,c>0$ and $a+b+c=1$ show that $6+3\left(\frac{ab}{c} + \frac{bc}{a} + \frac{ca}{b} \right) \le\frac{1}{a} + \frac{1}{b}+\frac{1}{c}$ My work : $6abc +3((ab)^2 + (bc)^2 +(ca)^2 ) \le ab+bc+...
the_bot_unknown's user avatar
0 votes
0 answers
17 views

How to reformulate optimization under equality constraint as a minimax optimization problem?

Suppose I have the following optimization under equality constraint: $\max_x f(x)$ subject to $h(x)=0$. How can I reformulate it as a minimax optimization problem in the sense of $\min_t \max_s Q(s,t)$...
ExcitedSnail's user avatar
0 votes
1 answer
57 views

Proving an inequality given a constraint

Given $a,b,c >0 $ and $ab+bc+ca=3$ prove the following inequality : $\large 3\left( \frac1{a} + \frac1{b} +\frac1{c} \right) \geqslant 6 + \frac{ab}{c} +\frac{bc}{a} + \frac{ca}{b}$ My work : LHS$= ...
the_bot_unknown's user avatar
4 votes
1 answer
378 views

Maximizing with Cauchy-Schwarz inequality

I want maximize the function $f(x)=\cos(x)+\sin(x)\cdot\cos(x)$, with $x \in (0,\frac{\pi}{2})$. By derivation $f'(x)=0 \Rightarrow x=\frac{\pi}{6}$. But, if we write $f(x)=\cos(x)+\frac{1}{2}\sin(2x)$...
Cgomes's user avatar
  • 1,258
0 votes
1 answer
34 views

Maxima-Minima problem in a chemical process

The problem goes like this- A chemical process is used to remove impurities from a pulp by bleaching process generating toxic and non-toxic wastes. The toxic waste generated is 0.05 times the square ...
Ash's user avatar
  • 21
4 votes
1 answer
82 views

Minimum value mismatch

Say we want to minimize the sum $S = 9 + k^{2}$ which is clearly $9$. But We know $(k-3)^{2} \geq 0$ Which gives $k^{2} + 9 \geq 6k$, so Minimum value of $S$ is $6k$ and clearly the equality holds ...
Anshu Gupta's user avatar
3 votes
1 answer
189 views

How would you find the maximum of this function? $\sin(\frac{1}{x-1})+\sin(\frac{1}{x+1})$ [closed]

Like the title says, I am trying to find the maximum of this function, which seems to be near 0.787. This is to "normalize" the function, so that the maximum is 1, so the x value is what I ...
nnabahi's user avatar
  • 83
0 votes
0 answers
48 views

Is there a simple expression for $\mathbb{E}[\max(a_1+b_1X_1, \ldots, a_n+b_nX_n)]$

Let us consider $X_1$, ..., $X_n$ iid with mean $0$ and variance $1$. Is there a simple expression for $\mathbb{E}[\max(a_1+b_1X_1, \ldots, a_n+b_nX_n)]$? Are there specific distributions for which ...
Ernest's user avatar
  • 29
-1 votes
1 answer
78 views

A min-max optimization problem for a given probability density function?

I am trying to solve a min-max optimization problem for a given probability density function. The problem is defined as follows: Let the random variable $X_\theta$ have the PDF: $$ f_\theta(x) = 1 - |...
Luna Belle's user avatar
0 votes
1 answer
35 views

Comparing a function and its derivative

Let $f : \mathbb{R} \rightarrow \mathbb{R} $ be defined by $$ f(x)= \begin{cases} (1-x)^2\sin(x^2) & \text{if } x\in (0,1) , \\ 0 & \text{otherwise} \end{cases} $$ And $f'$ be its ...
user-492177's user avatar
  • 2,589
1 vote
1 answer
17 views

Is $\inf_{a \in A}\sup_{b \in B} \|a-b\|_H = \sup_{b \in B} \inf_{a \in A}\|a-b\|_H$?

Let $A \subset H$ be a closed and convex non-empty subset of a Hilbert space $H$. Let $B \subset H$ be bounded and non-empty (if necessarily closed as well). Is it true that $$\inf_{a \in A}\sup_{b \...
math_guy's user avatar
  • 465
0 votes
1 answer
156 views

Using a cubic to predict a minimum between two points and their derivatives.

Background After fitting a parabola through a few samples (value + derivative), our algorithm - in search of a minimum - normally would jump to the vertex of that parabola. However, it is possible ...
Carlo Wood's user avatar
1 vote
1 answer
34 views

Can I take the derivative of only part of a function when solving for its maximum or minimum?

Let me illustrate my point with a simple example. Suppose we want to find the maximum of $f(x)=g(x)+h(x)$ (assuming all functions are differentiable), where differentiating $g(x)$ is straightforward ...
J H's user avatar
  • 13
3 votes
2 answers
101 views

Lagrange Multiplier Problem Leads to Tricky System of Equations

I am a little stumped on the following problem. Problem: Minimize the value of $108a + 27b^2 + 4c^3 + d^4$ subject to the constraint $ab + bc + cd + da = 25$. I have attempted to solve this using the ...
kjamesxyz's user avatar
4 votes
1 answer
124 views

Maximum value of $G = (x^3+y^3+z^3)\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)$

Inspired by this interesting minimum value problem that I solved and got $\text{minimum } G=41$, I was curious to know what would be the maximum value of $G$. The problem can be stated as follows: Let ...
Pustam Raut's user avatar
  • 2,322
-1 votes
1 answer
74 views

Why isn't there a local maximum too (constrained optimisation in $\mathbb{R}^3$)

It's perhaps a stupid question, but I'm dealing with this problem: I have to find max and or min of $f(x, y, z) = 2x^2 + y^2 - z^2$ constrained by $x+2y+z = 1$. Now I solved the problem and I found $x ...
Heidegger's user avatar
  • 3,492
2 votes
0 answers
38 views

Prove the area of a quadrilateral $A_1B_1C_1D_1$ has a local minimum at $A_1=A$

$ABCD$ is a cyclic quadrilateral. Its diagonals $AC,BD$ intersect at $P$. Let $E$ be the point on $AB$ such that $AE:EB=\tan\angle BAP:\tan\angle ABP$. Let $F$ be the point on $BC$ such that $BF:FC=\...
hbghlyj's user avatar
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