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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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maximizing concave function with parameter

We want to maximize the convex function: $$\vec{q} \cdot \vec{x} - \lambda||\vec{x} - \vec{1}||_2^2$$ where $\lambda$ is some parameter. I'm looking at a solution that states that the maximum is a ...
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How to solve this system of equations systematically?

This might seem a trivial problem, but I have some trouble in arranging the data. So suppose you are given $f(x,y)=x^2y^2(1+x+2y)$ and you want to find it's critical points. Thus we find $$\frac{\...
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How can we find some radius of circle which fully contains $x\arctan(x)-ax+y\arctan(y)-by=0$?

How can we find some radius of circle with center at origin which contains $x\arctan(x)-ax+y\arctan(y)-by=0$, where $\pi/2>a>0$ and $\pi/2>b>0$. I'm not sure how can we prove that ...
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Exercice using the derivation of the Euler-Lagrange equation

Here is a exercice using the derivation of the Euler-Lagrange equation: Here is the exercice: For a given function $f(x,u,u')$ and constants $K_1, K_2$ minimize the functional using the Euler-...
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How to study the critical points of a $2$-variable function?

I am revising some past exam questions and there is one that states: Study the critical points of the function: $$f(x,y)=x^2+y^2-x^4-y^4-2x^2y^2.$$ According to my professor, this is what I have ...
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Find the minimum value of $\sqrt {2x^2+2y^2} +\sqrt {y^2+x^2-4y+4} +\sqrt {x^2+y^2-4x-4y+8}$

Given that $0\lt x\lt 2$ and $0\lt y\lt 2$ then find the minimum value of $$\sqrt {2x^2+2y^2} +\sqrt {y^2+x^2-4y+4} +\sqrt {x^2+y^2-4x-4y+8}$$ My try: On factorisation we need minimum value of $$\...
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How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$?

How can we find minimum radius of circle which contains $\arctan^2(x)+\arctan^2(y)=a$, where I think $a<\pi/2$? For example I have some plots from WolframAlpha and I see it depends on $a$. But ...
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3answers
57 views

Cleverly finding the minimum of function

I want to cleverly find minimum of function $$f(x,y)=a(y-x)^2+bx^2$$ when $$x^2+y^2=1.$$ By cleverly I mean by using some smart inequalities which give lower estimate for $f,$ and which become ...
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Find Minimum value of $\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$

Find Minimum value of $$f(x)=\sqrt{58-42x}+\sqrt{149-140\sqrt{1-x^2}}$$ My try: the domain of the function is $x \in [-1 \,\,\,1]$ Differentiating and equating it to zero we get $$f'(x)=\frac{-21}{\...
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Inconsistent lagrange multiplier

so I have a function $f = 2\pi r h$ with $r, h$ as incognites. I want to minimize it. The restriction $g = \pi r^2 h-0.25$ The problema is that when I do the method I get an inconsistency like: $...
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1answer
25 views

Minimum distance of curve from origin

I have a parabola $(y+5)^2 = 4x$ and I need to find its minimum distance from origin. Scientific calculators aren't allowed. I have tried : 1) Substituting parametric coordinates $(r\cos Q, r\sin Q)$...
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Most efficient method to find peak frequency of an FFT

I'm using a real to complex fft to get the peak frequency of a signal. What I want to know is the most efficient way to find the index of that peak. I currently use im^2 + re^2 and compare to the ...
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Max of conditional Negative Binomial

Suppose $X|K = w$ is a Negative Binomial with parameters $r$ and $q$. K follows a Binomial Distribution with parameters $m$ and $p$. I want to calculate the expected value of $$Z = max(X_1, X_2,...,...
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Calculus: Maxima and Minima Triangle Problems [closed]

A right triangle has hypotenuse of length 13 and one leg of length 5. Find the dimensions of the rectangle of largest area which has one side along the hypotenuse and the ends of the opposite side on ...
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How can we find some radius of circle so $-x\arctan(x)+0.2x-y\arctan(y)+0.9y=0$ will be fully inside this circle?

If he have this region $$ \begin{align} \ -x\arctan(x)+0.2x-y\arctan(y)+0.9y=0\\ \end{align} $$ How can we find some $R$ (maybe minimum) so this region will fully inside this circle $(x-a)^2+(y-b)^2=...
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Constraint Qualification in Lagrange

$f(x,y)=x^2-y^2$ subject to single constraint $g(x,y)=1-x-y=0$ For this question, I understand that the Constraint Qualification holds, since, rank of $D(g(x,y))=1$ everywhere. Solving Lagrange would ...
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Limits and continuous functions with Graph

Graph is given: $\lim_{x→2^-} f(x)$ $\lim_{x→2^+} f(x)$ At $x=2$, is the function continuous from the left or continuous from the right? $\varliminf_{x→2} f(x)$ (lim inferior) $\varlimsup_{x→2} f(x)$ ...
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208 views

Real Analysis - Continuity

a. Give an example of a function defined everywhere on the interval $[0,1]$, which does not achieve its maximum. b. Give an example of a function defined on $\mathbb{R}$, that is nowhere continuous. ...
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Combinatorics: How many people speak German, French, English and Portuguese?

In a group of $32$ persons, $20$ speak German, $16$ speak French, $26$ speak English, and $16$ speak Portuguese. Every person in this group speaks at least one of these four languages. How can I ...
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1answer
77 views

Maximum of $ab+2bc+3ca$ with $a^4+b^4+c^4=1$

Let $a,b,c\in \mathbb R^+$ with $a^4+b^4+c^4=1$. What is the maximal value $ab+2bc+3ca$ can take? I tried using Cauchy-Schwarz several different ways and the best upper bound I got was $\sqrt{14}$, ...
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maxima/minima $h(v,w,x,y) := 6v^2-12v+arctan(w)- \frac{1}{2}w+\exp(x^2)+x^2+y^2+\frac{1}{4}xy$

Let $h: \{(v,w,x,y) \in \mathbb{R}^4 : w <0 \} \to \mathbb{R}$ with $h(v,w,x,y) := 6v^2-12v+arctan(w)- \frac{1}{2}w+\exp(x^2)+x^2+y^2+\frac{1}{4}xy$ How can one find the criticial points, i.e. the ...
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2answers
33 views

Point $x \in \mathbb{R}^n$ that minimizes sum of distance squares $\sum_{\mathcal{l}=1}^{k} \Vert x-a^{\mathcal{(l)}} \Vert _2^2$

Let $a^{(1)},...,a^{(k)} \in \mathbb{R}^n$. How can one find the point $x \in \mathbb{R}^n$, which minimizes the sum of distance squares $\sum_{\mathcal{l}=1}^{k} \Vert x-a^{\mathcal{(l)}} \Vert _2^2$...
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1answer
19 views

Finding the maximum of a function on a specific interval

I have a problem at it is as follows. I've to find the maxium value of the following function between to time points, namely $t=0$ and $t=\frac{2}{50}$. The function is the following: $$8-\exp(-\frac{...
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3answers
42 views

Solving the following maximization problem analytically?

Is it possible to solve the value of lambda that maximizes the following equation analytically?: $$ \frac{1-e^{-30 \lambda}}{30 \lambda} - e^{-30 \lambda} $$, So then the derivative is equal to: $$...
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Does $f(x,y,z)=(1 + |x|+|y|) \cosh(|x|-|y| + z^2)$ have extrema other than at the origin?

Let $f(x,y,z)=(1 + |x|+|y|) \cosh(|x|-|y| + z^2)$. $f(0,0,0)=1<f(x,y,z)$ for different $x,y,z$ so it is a global minimum. Using $f(x,y,z)=f(|x|,|y|,|z|)$ I found there are no extrema at points ...
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How is it possible to minimize a definite integral, if there are three parameters that can be varied in the integrand?

I have been attempting to minimize the following integral: $$ T=\int_{a}^{b}\sqrt{\frac{1+y'^2}{2gy}}dx, $$ knowing that $y$ is of the form: $$ y = a_0 +a_1 x+a_2x^2 $$ and that $y(0)=2$ and $y(\pi)...
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1answer
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How to determine local extrema for $f(x) = x\cdot \sin(x) ^ {\sin(x)}$

I need to find the local extrema points of the following function: $f(x) = x\cdot\sin(x) ^ {\sin(x)}$ I was already able to derive to this function: $f'(x) = x (\ln(\sin(x))+1)\cos(x)\sin(x)^{\sin(...
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3answers
78 views

Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$

Minimize $x^2+6y^2+4z^2$ subject to $x+2y+z-4=0$ and $2x^2+y^2=16$ My try: By Lagrange Multiplier method we have $$L(x,y,z,\lambda, \mu)=(x^2+6y^2+4z^2)+\lambda(x+2y+z-4)+\mu(2x^2+y^2-16)$$ For $$...
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2answers
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A Problem of Lagrange Multiplier

The problem is find the minimum value of $x^2+y^2+z^2$ subject to the condition $x+y+z=1$ and $xyz+1=0$. Let $f(x,y,z)=x^2+y^2+z^2$, then after some calculation I got this two equations: $...
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5answers
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State if the statement is True or False: The maximum value of $2x^3-9x^2-24x-20$ is $-7$.

State if the statement is True or False: The maximum value of $2x^3-9x^2-24x-20$ is $-7$. Let $f(x) = 2x^3-9x^2-24x-20$. If we go by the derivative test: $$f'(x) = 6x^2-18x-24 \ \ \& \ \ f'(x)...
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Maximum of a two variable function within a defined domain

The function is this: $xye^{\frac{(x+y^2)}{4}}$ in the domain $x+y\geq1, y\geq0$ I have done the partial derivatives to get the stationary point $(0,0)$ but it is not a maximum, it is a saddle point. ...
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0answers
35 views

Minimisation of integral including absolute values

I need to solve the following $$\min_{c,d} \int_0^1 |ct+d-t^2| dt.$$ Since the integrand is of degree two, I considered to split the integral in three. My problem however, is that the bounds would ...
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1answer
42 views

Using the Hessian Matrix to classify points

From what I've gathered from my calculus supplements and the web, I want to know if I have the general computation procedure understood correctly. Example: Given f such that f(x,y) = ___. Find and ...
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Minimize trace of $A$ given that $A−N$ is positive semi-definite and $A$ is diagonal

\begin{array}{ll} \text{minimize} & \mbox{tr} (\mathrm A)\\ \text{subject to} & \mathrm A - \mathrm N \succeq \mathrm O_n\end{array} where $A$ and $N$ are pd matrices, and $A$ is diagonal. ...
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Maximizing the trace of product of matrices under fixed spectrum

Is it correct that under fixed spectrum, $\operatorname{tr}(AB)$ is maximized when $A$ and $B$ share the same eigenbasis? If yes, how can this be shown?
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Determine square of distance between local maxima and local minima [duplicate]

P(x) is a polynomial of degree three having local maxima at x = -1 and given P(-1) = 10, P(1) = -6. Further P'(x) has local minima at x = 1. Determine square of distance between local maxima and local ...
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1answer
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Proving that cuboid of maximum volume in a sphere is a cube.

I was preparing for my maths test . And preparing application of derivative (theory based question ) there I saw a problem of proving rectangle of maximum area in a circle is square . So there were ...
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1answer
22 views

Finding the Shape of a Graph Using the Min and Max

QUESTION: Let $f(x)=2x^{2}-2x^{4}$. Find the open intervals on which $f$ is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). $f$ is increasing on the ...
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2answers
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Maximum length perimeter of a box whose diagonal is 10 unit long.

Suppose the diagonal of a three-dimentional box has length $10$, what is its maximum perimeter length? Here is my solution: Let the three edges of the box adjacent to a vertex be labelled $a,b,c$. ...
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1answer
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making sure we found all the extremals

I am curios to know whether there is anyway to be sure that we found all the stationary points using Lagrange multiplier method.? Thank you.
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1answer
35 views

Finding the maximum and minimum of $2x-y-5z=k$

Problem: Find the maximum and minimum of $2x-y-5z$ about $$ x,y,z \in \mathbb R$$ that satisfy Conditions $$x^2+y^2+z^2=9$$ $$x-y-z=1$$ $$2x+y+2z\ge0$$ I can solve the problem without the third ...
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1answer
38 views

Prove that Dual linear program does not have finite optimal solution

Consider the following $\displaystyle \max z=x+2y\\s.t.-x+y\le-2\\4x+y\le4\\ x,y\ge0$ Find the dual program and prove graphically that D has no finite optimal solution. Solution The dual is given ...
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1answer
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Finding maximum and minimum values of $\sin(2x) - x$ for $x\in\left[-\frac \pi 2, \frac \pi 2\right]$

This is a problem that I wasn't able to solve. Please help. Find the maximum and minimum values of the function: $$y = \sin(2x) - x$$ where domain of $x$ is $\left[-\frac \pi 2, \frac \pi 2\...
2
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1answer
34 views

How to form a set of numbers whose product is maximum?

Consider this example of constructing two numbers by concatenating the numbers of the set $S = \{0,1,2,\ldots,9\}$ without repetition in such a way that the product of the two numbers formed is ...
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0answers
30 views

Solve ${W_{-1}}'(-1/x)=-1$

This problem arose when I tried to find the maximum of $f(x)=\ln(x\ln(x\ln(x\cdots)))-x$. This can be written as $f(x)=\exp(-W_{-1}(-1/x))/x -x$ by substituting the recursion into $f$. The negative ...
1
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1answer
50 views

Prove that if f is continous on R and the limit as x approaches infinity and negative infinity is infinity, then f obtains its minimum value in R? [duplicate]

Problem: Suppose $f$ is a continous function defined on $\mathbb{R}$ s.t. $\lim\limits_{x\to -\infty}= \lim\limits_{x\to\infty} = \infty$. Then $f$ obtains its minimum value for some $x\in\mathbb{R}$....
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0answers
21 views

How can I solve this system of equations (Lagrange multiplier problem)?

A rectangular box with no top is to have a surface area of 16$m^2$. Find the dimensions that maximize it's volume. Here is what I have: Objective function: $f(x)=xyz$ Constraint function: $2xz+2yz+...
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0answers
27 views

Minimalization of one variable function

I am trying to solve the problem of finding a point, which minimizes the time required to move from some point to another, some of the time swimming, some running, anyways in order to do that i have ...
2
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1answer
85 views

Are $f(x,y) :=$ min{x,y} and $g:\mathbb{R^n} \to \mathbb{R}$ with $g(x) := \left\lVert x \right\rVert_2$ partially differentiable?

I have to find out if the function $f: \mathbb{R^2} \to \mathbb{R}$ with $f(x,y) :=$ min{x,y} and the function $g:\mathbb{R^n} \to \mathbb{R}$ with $g(x) := \left\lVert x \right\rVert_2$ are ...
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0answers
8 views

Related Rates: Spheres

A certain magical substance that is used to make solid magical spheres costs \$800 per cubic foot. The power of a magical sphere depends on its surface area, and a magical sphere can be sold for \$60 ...