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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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40 views

Minimun of a function that involves a defined integral

Let be $a,b\in \mathbb{R}$ with $a<b$ and consider a continuous and strictly increasing function $f:[a,b]\to \mathbb{R}$. For $y\in \mathbb{R}$ let define, $$E(y)=\int_a^{b}|f(x)-y|dx.$$Show that $...
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37 views

Maximum and minimum of $\frac{1}{n} \cot(n \pi \phi)$, $\phi$ Golden ratio

Studying aspects of the problem https://math.stackexchange.com/a/3186019/198592 I stumbled on this question. Designating the golden ration by $\phi=\frac{1+\sqrt{5}}{2} \simeq 1.61803$ and letting $a(...
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Characterizing class of functions from their gradient descent trajectory

Let $f : \mathbb R^d \to \mathbb R$ be a function with at least one minimum. Suppose that $f$ has the property that the gradient descent trajectory from any point of the function is a straight line to ...
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Find Maximum of any discrete function (not necessarily a PDF)

How can we find the maximum of any discrete function, say $$ f(n)=\frac{(n+1)^2}{2^n},\quad n\in \mathbb{N} $$ that is not the PDF of any distribution? (This query is unrelated to statistics.) By ...
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4answers
37 views

Minimum value of $y=\sin( 2x) - x$, where $x\in [-\frac{\pi}2,\frac{\pi}2]$

I tried applying the concept that at minima, derivative of $y$ with respect to $x$ should be zero, but realised that it fails as the domain is restricted. Rightly, upon plotting the graph, we can see ...
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2answers
37 views

Range of $f(x)$ when $e^x + e^{f(x)} = e$. [closed]

How to find range of the function $f(x)$ in the equation below : $$ e^x + e^{f(x)} = e $$
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20 views

Find the range of the following equation

Find the range of function $$f: [0,1]\to\mathbb{R}, f(x) = x^3-x^2+4x+2\sin^{-1}x$$ I like to know calculus solution if possible.
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1answer
24 views

maximum of the function $f(x) = \sum_{n\geq 1 } \sin n x /n^\alpha $

Here $\alpha >1 $. The function is defined as $$f(x) = \sum_{n\geq 1 } \sin n x /n^\alpha .$$ The domain is $(0, \pi)$. We know that if $\alpha = 1$, $f(x) = (\pi -x )/2$. If $\alpha >1$, ...
3
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1answer
48 views

How to solve this minimization problem with the power $0<p<1$?

Suppose that $0<p<1$, how to solve the problem $$ \min_{0\leq m \leq 1} 99m^p + (1-m)^p $$ I know the solution is $m^\star = 0$ and it also can be verified easily by a plot, but I can not prove ...
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2answers
79 views

maximum value of $\sum (a-b)^2$

If $a^2+b^2+c^2=5$ and $a,b,c \in \mathbb{R},$ find the maximum value of $(a-b)^2+(b-c)^2+(c-a)^2$. My Try: $(a-b)^2+(b-c)^2+(c-a)^2=2(a^2+b^2+c^2)-2(ab+bc+ac)$ $$=10-2(ab+bc+ac)$$ Now this ...
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2answers
35 views

Minimizing lengths of cevians in an isosceles right triangle

Consider isosceles right triangle $ABC$ with $BC$ as the hypotenuse and $AB=AC=6$. $E$ is on $BC$ and $F$ is on $AB$ such that $AE+EF+FC$ is minimized. Compute $EF$. My thought process: I reflected ...
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2answers
69 views

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Let $a,b,c$ be non-negative real numbers such that $c \geq 1$ and that $a+b+c=2$. Find the minimum and maximum values of $$P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$$ To find the minimum of $P$ ...
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38 views

How to find intersecting point of following type of functions?

We are given a function,$$f(x,y)=4x^2-xy+4y^2+x^3y+xy^3-4$$ Now to calculate minimum and maximum value of $f(x,y)$, i first calculated $f_x$ and $f_y$ for stationary points. which gave $$f_x=8x-y+3x^...
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1answer
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Finding points which are not local maximum or minimum

Consider the picture below: This is the levels curves for a function $f(x,y)$ where: Blue line is the partial derivative of $f(x,y)$ with respect to x Red line is the partial derivative of $f(x,y)$ ...
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1answer
62 views

Minimise a function of 3 variables.

If I have a function of 3 positive variables, $k$, $m$ and $n$, how can I find what value of $k$ minimises this function (in terms of $m$ and $n$)? The particular function I'm interested in is this: ...
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1answer
23 views

greatest value of function depends on parameter $k$

Find the greatest value of the function $f(x)=x^4-6kx^2+k^2$on the interval $[-2,1]$ depending on the parameter $k$ My Try: $$f(x)=x^4-6kx^2+9k^2-8k^2$$ $$f(x)=(x^2-k)^2-8k^2$$ from $-2 \...
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2answers
66 views

Find the minimum of $\space\frac{1}{x}+\frac{1}{y}+c\cdot xy\space$ subject to $\space x+y-c=0$

Let $f(x,y):\mathbb{D}\rightarrow\mathbb{R}$ be the function: $$f(x,y)=\frac{1}{x}+\frac{1}{y}+c\cdot xy\space\space|\space\space c\in(0,\sqrt[4]8)\text{ $\space$constant}$$ $$\mathbb{D}=\{(x,y)\...
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0answers
61 views

the suprema of $p$ norms in a $n$-ball

CONTEXT I'm writing a proof. I'm trying to make sure my mathematical writing is correct. I want verify that the supremum $$\sup\left\{\left\|\textbf{x}\right\|_p \mid \textbf{x} \in B_r{\left(\bf{...
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2answers
49 views

Area of a rectangle inside a triangle with given coordinates

Given a triangle with vertices at points $(0, -a), (0, a), (b, 0)$, where $a > 0$, find the maximal area and the dimensions (base and height) of a rectangle that can be contained within the ...
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15 views

A question from Calculus (Morris Kline). I don't understand the solution in the solution manual

a store is to be built with a rectangular floor area of 10000 square feet. The front wall will cost twice as much per linear foot as the side and back walls. The height of the store is to be 10 feet. ...
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1answer
35 views

Find the minimum value of $\sec 2A+\sec 2B$, where $A + B$ is constant. $A$ and $B$ belong to $(0,π/4)$ using graph of sec x.

Find the minimum value of $\sec 2A+\sec 2B$, where $A + B$ is constant. $A$ and $B$ belong to $(0,π/4)$ using graph. Obviously one can solve this question using Lagrange multipliers or find maxima-...
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2answers
46 views

Classifying the stationary points of $f(x, y) = 4xy-x^4-y^4 $

$f(x, y) = 4xy-x^4-y^4 $ The gradient of this function is $0$ in $(-1, -1), (0, 0),(1, 1)$ I tried to compute the determinant of the Hessian Matrix, but it's 0 for every point, I always get a null ...
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0answers
16 views

Why use chebyshev polynomial in this problem?

$f(x)$ is polynomial degree of 6. For $-1=<x=<1$ , $0=<f(x)=<1$ . What is maximum value of leading coefficient of $f(x)$. I saw solution, solution claim $g(x)=2f(x) -1$ and $g(x)=T_6 (x)$...
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3answers
28 views

decide if 3 variables function has a minimum and maximum value

There is a question in my textbook that says decide if the function $$f(x,y,z) = 2x^3+2y^3+2z^3-3xy-3yz-3zx$$ has a minimum and maximum value on R^3 And it says that the solution is to look at ...
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1answer
37 views

Uniform convergence of $(1-\frac{x}{n})^n$

Let $f_n : \mathbb{R}^+ \to \mathbb{R} : x \mapsto (1-\frac{x}{n})^n \mathbb{1}_{[0,n]}$. I would like to prove that the sequence $(f_n)_n$ converge uniformly on $\mathbb{R}^+$to $x \mapsto e^{-x}$ ...
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Intuitive reason that $(1/n)^n$ is maximized for $n = 1/e$

Consider the function $f(n) = \Big( \dfrac{1}{n} \Big)^n$. By setting $f'(n) = 0$, we find that the maximum of $f(n)$ occurs at $n = \dfrac{1}{e}$. Going through the calculations, there doesn't seem ...
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2answers
33 views

Solving for global Maximum and minimum on a interval

We need to determine the global maximum and minimum of: $f(x,y)=y^2-16x^2$ on the interval of: $\{(x,y) : y ≤ 1−x^2,y ≥ 0\}$ My initial thought was that I could use extreme value theorem, later ...
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1answer
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Multivariable calculus - Trying to understand if a stationary point is a saddle point, max or min

I have the following function $f(x, y) = x^4+y^4 $ $(0, 0)$ is a stationary point, so I calculat the determinant of the Hessian Matrix, which is 0, so I try to understand what kind of point that is ...
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1answer
16 views

Saddle point, point of inflection, extremum, stationary point

What is the difference between a point of inflection and a saddle point? What is the difference between an extremum and a stationary point?
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1answer
29 views

Minimizing sequence of functional on $L^p$ space

Let $\Omega$ be a Polish space with its Borel $\sigma$ algebra and a non atomic probability measure. Let $$U : L^2(\Omega, \mathbb{R}^d) \to \mathbb{R}$$ be a $C^1$ function (in Frechèt sense). Let $...
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0answers
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Optimizing quadratic function integrated over distribution

I got the following problem. I have to optimize a set of functions that are of the form $F_k(x,c_i,f) = \sum_{i,j} c_i c_j \int_0^{y_{MAX}} dy f(y) M_{i,j,k}(x,y)$ where $f(y)$ is a normalised ...
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4answers
97 views

$x^2 + y^2+xy = 1$ , then find the minimum of $x^3 y + xy^3 +4$

x and y belongs to real numbers. $ x^2 + y^2+xy = 1 $. then find the minimum value of $x^3 y + xy^3 +4$. I assume $ x = r \sin (w)$ and $ y = r\cos(w) $. $ x^3 y + xy^3 +4 = L $ which give me $ \...
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0answers
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What are spurious local minima?

I found this strange terminology "spurious Local Minima" in many research papers including https://arxiv.org/pdf/1712.00779.pdf Heuristically, I interpret this as a local minimum which does not ...
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1answer
22 views

Identity for multiple variables minimization

I would like the prove the following identity: $$\min\limits_{\mathbf{x},\mathbf{y}}[f(\mathbf{x})+g(\mathbf{y})]=\min\limits_{\mathbf{x}}[f(\mathbf{x})]+\min\limits_{\mathbf{y}}[g(\mathbf{y})]$$ At ...
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1answer
57 views

How to find the minimum of $f(x)=\frac{1}{\sqrt{x+1}}+\frac{\sqrt x}{\sqrt{x+p^2}}$? [closed]

Does anybody have any idea to calculate the minimum of $f(x)=\frac{1}{\sqrt{x+1}}+\frac{\sqrt x}{\sqrt{x+p^2}}$ without using the derivative of $f$?($p$ is a given constant)
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2answers
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Upper Bound for Polynomial Using Evenly Spaced Points

Suppose that $x \in [0,1]$ and the points $x_1, x_2,\ldots x_n$ are evenly spaced in the interval $[0,1]$. I am trying to find a tight bound for the maximum of: $$(x - x_1)(x-x_2)\cdots(x-x_n) $$ I ...
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4answers
114 views

What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$

Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ They all are positive terms so arithmetic mean is greater than equal to geometric mean $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x\geq 3( ...
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2answers
54 views

Find minimum perimeter of the triangle circumscribing semicircle

The following diagram shows triangle circumscribing a semi circle of unit radius. Find minimum perimeter of triangle My try: Letting $$AP=AQ=x$$ By power of a point we have: $$BP^2=OB^2-1$$ where $...
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1answer
24 views

Minimization of piecewise real multivariate function

I want to minimize a multivariable function with the following form: $$ f(\vec{x}) = g(\vec{x}) + | h (\vec{x}) | , $$ this is a piecewise function depending on the sign of $h (\vec{x})$, and I assume ...
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0answers
18 views

Inequality of $\min$ and $\max$ of functions and their convex-conjugates

I came across this problem where it asks me to prove the following inequality: $$\min\limits_{\mathbf{x}}[f(\mathbf{x})+g(\mathbf{x})]\geq\max\left(-\min\limits_{\mathbf{y}}[f^*(\mathbf{y})+g^*(\...
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4answers
123 views

minimum of $a^2+4b^2+c^2$ given $2a+b+3c=20$

If $a,b,c\in\mathbb{R}$ and $2a+b+3c=20.$ Then minimum value of $a^2+4b^2+c^2$ is what i try Cauchy schwarz inequality $$(a^2+(2b)^2+c^2)(2^2+\frac{1}{2^2}+3^2)\geq (2a+b+3c)^2$$ How do i solve ...
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0answers
34 views

A difficulty in understanding the solution of a problem.2

The problem and its solution is given here : A difficulty in understanding the solution of a problem. But I do not understand how $f_{x_{k}x_{k}}^{''} = -k(k + 1) c^{\frac{n^2 + n -2}{2}}$ and $f_{...
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2answers
51 views

Minimum value of PA+PB+AB

If $P(2,1) $ and $A $ and $B $ lie on $x$ axis and $y=x $ respectively, then find the minimum value of $PA+PB+AB $ . If $A$ was given , I could have worked geometrically, by using that image of $P$...
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1answer
17 views

Should the critical points belong to the domain of $f$?

If $f(x,y) = x ^2 + xy + y^2 - 4 \ln x - 10 \ln y,$ and I found the critical points to be {(1,2), (-1,-2),$(4/\sqrt{3}, -5/\sqrt{3})$, $(-4/\sqrt{3}, 5/\sqrt{3}) $} .... should I exclude the points ...
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2answers
30 views

finding the extreme values of $f$.

If $f(x,y) = x ^2 + xy + y^2 - 4 \ln x - 10 \ln y,$ and I found $f_{x}^{'} = 2x + y - 4/x$ which means, $$\tag{1}2x^2 + xy = 4$$ and also I found $f_{y}^{'} = 2y + x - 10/y$ which means, $$\tag{2}2y^2 ...
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0answers
68 views

Find max and min of $11\cos^2\theta+3\sin\theta+6\sin\theta\cos\theta+5$

I'm trying to solve the following question from my textbook. Using the identity $\cos2\theta=2\cos^2\theta-1$ and $a\cos\theta+b\sin\theta=\sqrt{a^2+b^2}\cos(\theta-c)$, where $c$ is a constant, ...
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0answers
29 views

Discussing the convergence of $\int_{-1}^1 dx/x^4$ [closed]

I broke the improper integral into: $$\lim_{a\to0} \int_{-1}^{a-0} \frac{dx}{x^4} +\lim_{a\to0} \int_{0+a}^1 \frac{dx}{x^4},$$ as $0$ is the point of discontinuity. Now, while plugging the limits of ...
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0answers
10 views

Union and min function

Given two sets of set of numbers $\{X_1,\ldots, X_N\}$ and $\{Y_1, \ldots, Y_N\}$, if $$\forall i, \min_{x \in X_i} x \leq \min_{y \in Y_i} y$$ Then, $$ \min_{x \in \cup_i X_i} x <= \min_{y \in \...
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1answer
35 views

Find all local maximum and minimum points of the function $f$.

Any help with this problem I have would be so much appreciated, i've been going round in circles and have no idea what I'm really doing. I have the problem: Find all local maximum and minimum points ...
1
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2answers
63 views

Least value of $\left|2017 z+\frac{1}{2018z}\right|$ for complex $z$ with $|z|\geq 2019$

Let $z=x+iy$ be a complex number with $|z|\geq 2019$. What is the least value of $$\left|2017 z+\frac{1}{2018z}\right| \, ?$$ What I tried: Put $z=re^{i\alpha}$. Then $z^{-1}=\frac{1}{r}e^{-i\alpha}$ ...