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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 ...

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How to know if a probabilistic function is increasing or decreasing without actually computing it?

Problem Setup Consider the following optimization problem \begin{equation*} \begin{aligned} & \underset{P_{a},\lambda_{a}}{\text{maximize}} & & P_{cov}(P_{a},\lambda_{a}) \\ & \text{...
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Creating a graph given condition of local minima and maxima

I had a question that goes like this: Let $m$ be the number of local minima and $M$ be the number of local maxima. Can you create a function where $M > m + 2$ ? Graph. I tried graphing it using ...
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is true that $\int_0^1 [f(x)]^n g(x) \: dx = a_n \: \forall \: 0<n \in \mathbb{Z} \implies \text{max} \: f \: \exists $

\begin{align} I_n = \int_0^1 [f(x)]^n g(x) \: dx = a_n \: \forall \: 0<n \in \mathbb{Z} \implies \text{max} \: f \: \exists \end{align} $f(x)$ and $g(x)$ are functions defined on the interval $(0,...
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Minimization of Floor and Ceiling Functions

So the problem at hand is: Find the minimum value of the following function for $ x> 0 $: $$ \def\lc{\left\lceil} \def\rc{\right\rceil} \newcommand{\floor}[1]{\lfloor #1 \rfloor} x\lc x \rc + ...
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AM GM inequality using conditional extrema

Given: $\prod_{i=1}^n x_i = 1$ leads to constraint function $G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$ ($\prod_{i=1}^n x_i =x_1 x_2...x_n$) Task is to to find the minimum using conditional extrema of ...
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38 views

Find maximum value

Given $0 \leq a,b,c \leq \dfrac{3}{2}$ satisfying $a+b+c=3$. Find the maximum value of $$N=a^3+b^3+c^3+4abc.$$ I think the equality does not occur when $a=b=c=1$ as usual. I get stuck in finding the ...
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Maximise $(x+1)\sqrt{1-x^2}$ without calculus

Problem Maximise $f:[-1,1]\rightarrow \mathbb{R}$, with $f(x)=(1+x)\sqrt{1-x^2}$ With calculus, this problem would be easily solved by setting $f'(x)=0$ and obtaining $x=\frac{1}{2}$, then checking ...
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Minimization of the distance between 2 vectors

Find the value of $t$ for which the vector ${v} = \begin{pmatrix} 2 \\ -3 \\ -3 \end{pmatrix} + \begin{pmatrix} 7 \\ 5 \\ -1 \end{pmatrix} t$ is closest to ${a} = \begin{pmatrix} 4 \\ 4 \\ 5 \end{...
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Maximum for function $(\theta-(\frac{\mu}{p-x})^a)·x$ [on hold]

Struggling with finding the maximum for this function: $$f(x)=\Bigl(\theta-\Bigl(\frac{\mu}{p-x}\Bigr)^a\Bigr)·x$$ where $\theta>0, \mu>0, p>0, \alpha>1$. Wolfram Alpha gives me the ...
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Calculas, finding the minimum no. of roots

If f(x) is a differentiable function for all real x such that, F(x) has fundamental period 2 F(x) =0 has exactly 2 solutions in<[0,2] F(0)≠0 Also consider g(x)=f(x)cosx and h(x)=g'(x) Then which ...
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Prove $f(x)=x^n(1-x)\lt \frac{1}{ne}$ for all $n\in \mathbb{N},x\in(0,1)$

Prove $f(x)=x^n(1-x)\lt \frac{1}{ne}$ for all $n\in \mathbb{N},x\in(0,1)$. My try: $f'(x)=nx^{n-1}-(n+1)x^n=x^{n-1}(n-(n+1)x)=0\Rightarrow x=\frac{n}{n+1} $, hence $\max_{x\in(0,1)} f(x)=f(\frac{n}{n+...
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If $x>0, y>0,x+y=\frac{\pi}{3}$ then maximum value of $\tan x\tan y$ [duplicate]

If $x>0, y>0$ and $x+y=\frac{\pi}{3}$, then find the maximum value of $\tan x\tan y$ My Attempt $x>0, y>0, x+y=\frac{\pi}{3}\implies x, y$ in $1^\text{st}$ quadrant. $\tan x, \tan y>...
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Derivative and extremum

I try to solve a problem but I just can't figure it out: I have this equation as a statement: $f(x)= x^4+ax^3+bx^2+cx+d$ There is a maximum at $x=0$ There are four propositions and one is supposed ...
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Prove $f(x)$ can reach its minimum value over $(a,+\infty).$

Problem If $f(x)$ is continuous over $(a,+\infty)$, and $\lim\limits_{x \to a+}f(x)=+\infty,$$\lim\limits_{x \to +\infty}f(x)=+\infty,$ then $f(x)$ can reach its minimum value over $(a,+\infty)$. ...
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Maxima and Minima of sinusoidal harmonics with unequal co-efficients

This question is related to a similar question I asked before but this time the equation has changed. Maxima and minima of sinusoidal function of harmonics The equation to which i am trying to find a ...
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27 views

Maxima and minima of sinusoidal function of harmonics

I have a function $$f(t)=\sin(5t)+\sin(7t)+\sin(9t)+\sin(11t)+\sin(13t).$$ I need to find the maxima and minima of this function. I know that using first and second derivative test one can find the ...
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Maximum and minimum absolute value of a complex number

Let, $z \in \mathbb C$ and $|z|=2$. What's the maximum and minimum value of $$\left| z - \frac{1}{z} \right|? $$ I only have a vague idea to attack this problem. Here's my thinking : Let $z=a+bi$ ...
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2answers
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Find value of n such the function has local mimima at x=1.

If $f$ is defined by $$f(x)=(x^2-1)^n(x^2+x-1)$$ then $f$ has a local minimum at $x=1$, when (i) $n=2$ (ii) $n=3$ (iii) $n=4$ (iv) $n=5$ Multiple options are correct. The given answer is $n=...
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maximising function subject to the constraint without using lagrange multiplier and other calculus techqniques

question: maximise the following function $f$ $$f= x^p y^q z^r$$ subject to the constraint $$ax+by+cz=p+q+r$$ i know how to do it using lagrange method of multipliers . but i'm looking for an ...
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argmax of sum of functions

Suppose I have two (finite) series of functions, $\{f_{k}\}_{k=1}^{K},\{g_{k}\}_{k=1}^{K}$. I know that: $$argmax_{x} f_{k}(x)=argmax_{x} g_{k}(x)$$ $$argmax_{x} f^{2}_{k}(x)=argmax_{x} g^{2}_{k}(x)$$ ...
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1answer
47 views

Second derivative test using $f_{yy}$ instead of $f_{xx}$?

The second derivative test for functions of two variables says to first find critical points. For each critical point one finds $$ D = f_{xx}f_{yy} - f_{xy}^2 $$ If $D>0$, the sign of $f_{xx}$ says ...
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2answers
35 views

Maximum between two consecutive minima, topology

I couldn't fall nasleep tonight due to a math problem I made up. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous fubction. Then, there is always a local maximum between two consecutive local minima. ...
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Determining the minimum value of the function $y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$

I am curious whether there is an algebraic verification for $y = x + 2\sqrt{x^2 - \sqrt{2}x + 1}$ having its minimum value of $\sqrt{2 + \sqrt{3}}$ at $\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$. I ...
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1answer
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How to detect inflection points without function expression in a live time series scenario?

Most of the learnings and examples out there talks about finding inflection point given there is a function expression. What happens when I do-not know the function expression? The movement of the ...
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1answer
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Multiple minima points and no maximum

Question: Use the bordered Hessian test to show that $f(x,y,z)=x^2+y^2+z^2$ under $g(x,y,z)=z-xy-2=0$ has two minimum points and no maximum (find the two points), and explain how this is possible. I'...
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1answer
38 views

$f(x,y)$ has a Hessian matrix is positive definite for all $(x,y)$, prove that $(a,b)$ is the unique absolute minimum for $f$?

suppose that $f(x,y) \in C^2$ with one critical point $(a,b)$ and that $f$ has the property that its Hessian matrix is positive definite for all $(x,y)$ except possibly at $(a,b)$ how can you prove ...
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Recommending value for $x$ which minimizes a function (only co-ordinates given)

I was asked this question by someone during an interview. Given a function $y = f(x)$, where $f(1) = 10$, $f(2) = 15$, $f(3) = 1$, $f(4) = 3$, $f(5) = 2$, $f(6) = 4$, $f(7) = 8$. What will be the ...
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1answer
24 views

Maximize $x(C-x)A^2+y(D-y)B^2+(x(D-y)+y(C-x))AB$ where $A,B,C,D$ are Constant

I am trying to figure out where the maximum occurs for the following expression of 2 variables ($x$ and $y$): $x(C-x)A^2+y(D-y)B^2+(x(D-y)+y(C-x))AB$ where $-1\leq A,B\leq 1$, $C\geq 2$ and $D\geq 2$...
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Minimum of a sum proof.

The problem I am working on is: Let $ Y = \{{y_1,y_2,y_3,...y_n\}}$ and $c=median(Y)$. Prove that: $$ \text{min}\left[\sum_{i=1}^n \lvert y_{i}-c\rvert\right]=c $$ My question is: Is the $\text{...
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1answer
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Distance of normal to the ellipse from the centre of the circle

Find the normals to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ which are farthest from the centre. My approach Let the point be $(x_1 ,y_1)$ and the tangent equation is $\frac{xx_1}{9}+\frac{yy_1}{...
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2answers
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Find the minimum value of $\sin^2\theta+\cos^2\theta+\csc^2\theta+\sec^2\theta+\tan^2\theta+\cot^2 \theta$ [duplicate]

What is the minimum value of this expression? $$\sin^2\theta+\cos^2\theta+\csc^2\theta+\sec^2\theta+\tan^2\theta+\cot^2 \theta$$ I tried grouping $\sin^2x+\csc^2x$, $\cos^2x+\sec^2x$ and $\cot^2x+\...
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Find Greatest length of log that can be floated

A Channel of $27$ metres wide is at right angles to channel of $64$ metres wide. Find Greatest length of log that can be floated in system of channels Is it not the Greatest length of log, the ...
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1answer
58 views

Can I minimize $x^a + y^a$ by minimizing $x+y$? Is there such a mathematical identity?

I would like to minimize the following expression: $$x^a+y^a.$$ I wonder if a mathematical identity exists where a minimization of $x+y$ implies a minimization of the above. Where: a is a positive ...
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1answer
29 views

Upper bound on Sum of square roots

Let $k_1,k_2\ldots k_t$ be integers and $\sum_{i=1}^{t}{k_i}=k$ where $k$ is fixed. What is the maximum value of $\sum_{i=1}^{t}\sqrt{k_i}$.
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Preservation of Minima with a Non-Linear Monotonic Mapping

I am trying to make a transformation on the set of parameters within the Ising model namely, $a_i$ and $b_{i,j}$. The Hamiltonian is: $H = \sum_{i} a_ix_i + \sum_i \sum_j b_{i,j}x_ix_j $ They need to ...
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52 views

A probability concerning the maximum and minimum of a simple random walk

Let $X_i$ be i.i.d. such that $\mathbb P(X_i = 1 )=\mathbb P(X_i=-1) =\frac1 2$. Let $a\in \{1,2,....\}$, now define the random walk, $S_0=a$ and $$S_n = a+\sum_{i=1}^n X_i$$ Now define the maximum ...
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1answer
119 views

Finding the minimal value of a $4\times 4$ determinant

The question. Let $\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb R^4$ be a vector with irrational coordinates. I am interested in finding the minimal value $\mu_\xi$ of $$\left\vert \det \begin{pmatrix} ...
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1answer
28 views

Can we add Lagrangian-like multipliers for a joint constraint $g(x, y)=0$ in a min-max problem?

Assume we have a problem $$ \min_x \max_y f(x, y) \ \ \text{ s.t. } \ g(x, y) = 0$$ I wonder if it is possible to incorporate this constraint into an objective in Lagrangian-like fashion: $$ \min_{...
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8answers
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The sum of two numbers is $28.$ Find the numbers if the sum of their squares is a minimum.

I have no clue where to start. The sum of two numbers is 28. Find the numbers if the sum of their squares is a minimum I am an eleventh-grader. I only learned how to find the minimum for functions ...
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2answers
46 views

About Finding Global Maximum and Proving its uniqueness

The problem is as shown. I tried using gradient and Hessian but can not make any conclusions from them. Any ideas? max $x_1^{a_1}x_2^{a_2}......x_n^{a_n}$ s.t. $\sum_{i=1}^nx_i=1, x_i>=0, i=1,2,.....
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0answers
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Ratio between the maximum and minimum of a random variable [closed]

Suppose we are drawing $n$ uniform random variables with replacement on $[a,b]$ where $1 \leq a \leq b < \infty$. What I need to argue that the maximum and minimum of such drawing will not be too ...
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1answer
39 views

Maximize a series

I have a set of positive numbers $X = \{x_1, x_2, x_3 \dots x_n \}$ such that $\sum x_i = m$. I am trying to maximize the following summation, $$ S = \sum_{x_i \in X, x_j\in X} (x_ix_j)^3 $$ I ...
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1answer
27 views

Clarification about the relation between maximization and minimization of objective functions

I am reading about how a maximization problem can be converted into a minimization problem: what I have understood is the following: ...
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1answer
30 views

Does a local max that is not strict imply a function is constant in some interval?

I have the following question. Thanks for any help in advance. Any hints would be appreciated. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ has a local max at x that is not a strict local maximum....
3
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3answers
75 views

Finding the minimum or maximum of a bivariate function when $f_{xx}\times f_{yy}-f_{xy}^2=0$.

I see that when $f_{xx}\times f_{yy}-f_{xy}^2<0$ then it is a saddle point. Also when $f_{xx}\times f_{yy}-f_{xy}^2>0$ then it is a minima or maxima. What is exactly happening when $f_{xx}\times ...
0
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2answers
34 views

Calculation of extrema points

Given is the function $f(x,y) = 3x^2y+4y^3-3x^2-12y^2+1$ I'm looking for the extrema points. Therefore I calculated $f_x(x,y)= 6xy-6x$ and $f_y(x,y)=3x^2+12y^2-24y$ and set them to zero to find the ...
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0answers
28 views

A question about Lagrange multiplier(when $\lambda=0$)

I need help in a maximization problem. where $R_s$ and $\Phi$ are $n$ by $1$, with other variables being scalars. The solution to this problem gives the following first order condition(FOC) i.e. ...
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1answer
33 views

max $\sin(x)/x$ without derivative

Showing $max(\frac{sin(x)}{x})=1$ is straight forward using l'hopital's rule. Is there another way to evaluate without using l'hopital's rule
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2answers
26 views

How to prove that maximum occurs at x=n/2 when n is even

I am stuck with this question: Using the recursion formula of a binomial distribution and for $\theta = \frac{1}{2}$: $b(x+1;n,\theta) = \frac{\theta(n-x)}{(x+1)(1-\theta)}*b(x;n,\theta)$ Show that ...
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0answers
11 views

Can we tell if a function has a max or min by looking along specific directions?

Suppose we have a smooth function $f$ from $\mathbb R^n\to\mathbb R$ such that $\nabla f(0)=0$, and we want to check if $f$ has a local maximum at $0$ (as opposed to a local min or a saddle point). ...