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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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Derivation of Maxima & Minima in two variables

For two variables i.e $\quad u=f(x,y)$: Condition for Maxima and Minima is given by: $$\frac{\partial f}{\partial x}=0 \quad , \quad \frac{\partial f}{\partial y}=0 \quad \implies \text{Critical ...
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A question about graph plotting.

Q. Plot $f(x)=\frac{x^2}{2}-\log(1+x^2)$ My approach: $$f ' (x) = x\big[1-\frac{2}{(1+x^2)}\big]$$ Equating the above with $0$; I get the maxima and minima points as $0,+1$ and $-1$. But when I ...
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1answer
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extrema(max-min) two variable constrained function $f(x,y)=x^2+y^2$ with “fixed” Hessian matrix

I have this function $f(x,y)=x^2+y^2$ under the constrain : $x^6+3y^2=1$ I use the Lagrange multiplier method : $$ \left\{ \begin{array}{} 2x=6x^5\lambda\\ 2y=6y\lambda\\ x^6+3y^2-1=0 \end{array} \...
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857 views

Minimum point of $x^2+y^2$ given that $x+y=10$

How do you I approach the following question: Find the smallest possible value of $x^2 + y^2$ given that $x + y = 10$. I can use my common sense and deduce that the minimum value is $5^2 + 5^2 = 50$....
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1answer
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The limit of the maximum of a sum of sines

I've recently stumbled upon the following problem from Brilliant: Compute the following: $$\lim_{n\to\infty}\max_{x\in[0,\pi]}\sum_{k=1}^n\frac{\sin(kx)}k$$ Options: $\...
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1answer
22 views

maximal point of a combination of continuous sub-additive monotone function

Let $f:\mathbb{R}\to \mathbb{R}$ be a monotonically increasing sub-additive continous function. Let $0<t\in \mathbb{R}$. Is $\frac{t}{2}$ a local maximum of the function $f\left( x \right) + f\left(...
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How to find the min and max values of a $y = \sin (2\pi t/23)$? [on hold]

I’d appreciate if you can help me about the formula of finding the $\max$ and $\min$ values of a $\sin$ graph: $y=\sin\left(\frac{2\pi t}{23}\right)$ $t$ represents the each day (in seconds) for the ...
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2answers
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Lagrange Multipliers: Absolute minimum of $f(x_1,x_2,…,x_n)$ on the boundary of the region $x^2_1+2x^2_2+3x^2_3+…+nx^2_n≤1$

I'm trying to use Lagrange Multipliers to solve the following problem Let $f(x_1,x_2,...,x_n)=x_1^2+x_2^2+...+x_n^2$ What is the absolute minimum of $f(x_1,x_2,...,x_n)$ on the boundary $x^2_1+2x^...
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1answer
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Derive the posterior mode

Consider random variable Y with a Poisson distribution: $$P(y|\theta) = \frac{\theta^y e^{-\theta}}{y!}, y=0,1,2,\ldots, \theta>0$$ Mean and variance of Y given $\theta$ are both equal to $\theta$. ...
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2answers
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Show $g:[0, 1] \rightarrow [0, 1]$ is continuous.

Let $f:[0, 1]×[0, 1] \rightarrow \Bbb{R}$ be continuous and assume that for all $x \in [0, 1]$ there is a unique $y_x$ such that $f(x,y_x)$ = $\max\{f(x,y):y∈[0, 1]\}$. If $g(x) =y_x$, then $g:[0, 1] \...
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1answer
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Prove That Area of Isoceles Triangle in an Ellipse is Maximum When Vertex On The Major Axis Lies On The Line Of Symmetry of the Triangle?

This is one of 101 classes questions whose solutions can be easily found on google, but most of the solutions assume without giving any proper line of reasoning that to maximize area (unique)vertex on ...
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1answer
28 views

How can the following non-convex problem be converted to a convex one? [closed]

Minimize: $x^2+xy+3y^2-x-4y+1$ Subject to: $(x-1)\log(1+\exp(x))\leq 0, x+y\geq0$
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1answer
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Bounding the dot product of two planar unit vectors.

Does there exist a continuous, monotone increasing function $f\colon[0,2]\to [0,1]$, satisfying $f(0)=0$ and $f(1)=1$, such that for all vectors $(a_1,b_1),(a_2,b_2)\in \mathbb{R}^2$ of unit length, i....
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Show that global minima exists or not

My function is defined $f(x,y) = 2x^3 + 3y^2+3x^2y-24y$. I found the critical points $(x,y) = (0,4), (-2,2)$ and $(4,-4)$. I also showed that $(0,4)$ is the local minimum by showing that the Hessian ...
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2answers
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Constrained minimization of multi-variable trigonometric function

Let f(x,y) = $\sin^2(x)$+$\arctan^3(xy)$. For every positive integer $n$ and every $M \ge 0$, find $$ \min_{(x,y)\in S_{n,M}} f(x,y), $$ where $S_{n,M} =[0,n \pi]×[−M,M]$,and determine where this ...
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1answer
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Finding the greatest value of $a^2b^3c^2$ if $a+b+c=3$ and all numbers are positive

Find the greatest value of $a^2b^3c^2$ if $a+b+c=3$ and all numbers are positive. Here is my attempt using $\text{AM-GM inequality}$: $$AM=\frac{a+b+c+a+b+c+b}{7}$$ $$GM=\sqrt[7]{a^2b^3c^2}$$ We ...
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2answers
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minimum value of $l$

Minimum positive real number $l$ for which $7\sqrt{a}+17\sqrt{b}+l\sqrt{c}\geq 2019.$ given that $a+b+c=1$ and $a,b,c>0$ what i try cauchy Inequality $$(7^2+17^2+l^2)(a+b+c)\geq \bigg(7\sqrt{...
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2answers
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Find maximum of $\cos(x)\cos(y)\cos(z)$ when $x + y + z = \frac{\pi}{2}$ and $x, y, z > 0$

Find maximum value of P = $\cos(x)\cos(y)\cos(z)$, given that $x + y + z = \frac{\pi}{2}$ and $x, y, z > 0$. Effort 1. I drew a quarter-circle, divided the square angle into three parts, and ...
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Optimization - Convergence to global minimum

Is it possible that a series of local maximum can converge to a global minimum? If it is possible, can you provide an example? If not, why is not possible? Thanks.
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3answers
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Two functions f(x)=x^x and f(x)=x^(-x)

I was playing with desmos. Then I found a pair of interesting graphs, namely $f(x)=x^x$(the purple one) and $f(x)=x^{-x}$.(the green one) They both have their global maximum/minimum at a point about $...
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3answers
64 views

maximum and minimum of $x^2+y^4$ for real $x,y$

If $y^2(y^2-6)+x^2-8x+24=0$ then maximum and minimum value of $x^2+y^4$ is what i try $y^4-6y^2+9+x^2-8x+16=1$ $(x-4)^2+(y^2-3)^2=1\cdots (1)$ How i find maximum and minimum of $x^2+y^4$ from $(1)...
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2answers
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Basic problem with local minimum and maximum.

Let's take a constant function for example f(x)=2 , x$\in$R. In every point of domain of f we have local minimum and maximum becouse : Let a $ \in $ R. Then exist $\alpha$ >0 that for every $x\in$B(a,$...
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7answers
92 views

Find the minimum value of $9x^2+2y^2-8xy-6x+11$

For any $x, y$ in real, find the minimum value of $9x^2+2y^2-8xy-6x+11$
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The greatest value of $|z|$ if $\Big|z+\frac{1}{z}\Big|=3$ where $z\in\mathbb{C}$

$\bigg|z+\dfrac{1}{z}\bigg|=3$ then the greatest value of $|z|$ is ___________ My Attempt $$ \bigg|z+\frac{1}{z}\bigg|=\bigg|\dfrac{z^2+1}{z}\bigg|=\frac{|z^2+1|}{|z|}=3\\ \bigg|z+\frac{1}{z}\bigg|=3\...
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1answer
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Maxima inside open set

Consider the function $$f(x,y) = e^xcos(y).$$ I am asked if this function has a maximum or a minimum inside the unit circle $\{x,y: x^2+y^2<1\}$. The answer provided here is that, since the ...
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1answer
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How to find minimum value

For $a=\sqrt{x^2-3\sqrt2x+9}$ and $b=\sqrt{x^2-5\sqrt2x+25}$ what is the value of $x$ when $a+b$ is minimum and how to find this? Thanks in advance.
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Function with a derivative with removable singularities, minimization problem

Consider $f(x) = \mid x - a \mid^3$ with $x \in [0,b], a \in (0,b)$. We all agree that the function is convex and it is minimized at $x=a$. $f(x)$ is simpler than the functions I actually have, whose ...
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5answers
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Maximizing $ (4a-3b)^2+(5b-4c)^2+(3c-5a)^2$, such that $a^2+b^2+c^2=1 $

If $$a^2+b^2+c^2=1 $$here a,b,c are the real numbers then find the maximum value of $$ (4a-3b)^2+(5b-4c)^2+(3c-5a)^2$$ I tried to think with vectors, that is direction cosines of lines. But then the ...
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1answer
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Prove if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function and $\lim_{x\to\pm\infty} f(x) = 0$, $f$ has a global maximum and minimum.

Prove if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function and $\lim_{x\to\pm\infty} f(x) = 0$, then $f$ has a global maximum and minimum. This is the exact question posed, but wouldn't a ...
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Effect of a convolution with a Bernoulli distribution on Rényi divergence

Let $P$ and $Q$ be two probability distributions on $\mathbb{Z}$. Let $D_\alpha(P\|Q)$ be the Rényi divergence of order $\alpha$ of $P$ and $Q$: $$ D_\alpha(P\|Q)=\frac{1}{\alpha-1}\sum_i\frac{P(i)^\...
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2answers
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Confusing Lagrange multipliers question

Let $a_1,a_2, \dots, a_n$ be reals, we define a function $f: \mathbb R^n \to \mathbb R$ by $f(x) = \sum_{i=1}^{n}a_ix_i-\sum_{i=1}^{n}x_i\ln(x_i)$, in addition, we are also given that $0 \cdot \ln(0)$ ...
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1answer
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Extremum value of $f(x,y)=ax^2+2hxy+by^2$ subject to constraints $g(x,y)=x^2+y^2-c^2=0$.

Find the extremum value of $f(x,y)=ax^2+2hxy+by^2$ subject to constraints $g(x,y)=x^2+y^2-c^2=0$, where $abc \neq 0$ and $(a+1)^2+4(a^2-b^2) \geq 0$. My attempt: I have use Lagrange Multipliers to ...
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1answer
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Let $a;b;c\in R+$ such that $a+\frac{b}{16}+\frac{c}{81}\le \:3;\frac{b}{16}+\frac{c}{81}\le 2;c\le 81$. Find maxima of $A$

Let $a;b;c\in R+$ such that $a+\frac{b}{16}+\frac{c}{81}\le \:3;\frac{b}{16}+\frac{c}{81}\le 2;c\le 81$. Find the maxima of function $$A=\sqrt[4]{a}+\sqrt[4]{b}+\sqrt[4]{c}$$ Wlog $a\le b\le c$ $f''(...
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1answer
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Calculating the minimum distance to the origin from a curve defined by $\frac{x^2}{4}+y^2+\frac{z^2}{4}=1$ and $x+y+z=1$

I want to calculate the points of the curve given by $$\frac{x^2}{4}+y^2+\frac{z^2}{4}=1,\qquad x+y+z=1$$ which are minimum and maximum distance to the origin. Using Lagrange multipliers, the maximum ...
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What is the minimum value of $f_\infty=\frac{x}{\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}}$?

In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers. What is the minimum value of $$f_\...
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0answers
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optimization exercise to find the maximum area of a rectangle formed by two rectangles with a line. Literal exercise.

A rectangle R in the plane has corners at (+-8, +-12), and a 100 by 100 square S is positioned in the plane so that its sides are paralleul to the coordinate axes and the lower left corner of S is on ...
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4answers
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Find a point $X$, in the plane of regular pentagon $ABCDE$, that minimizes $\frac{XA+XB}{XC+XD+XE}$.

Find such a point $X$, in the plane of the regular pentagon $ABCDE$, that the value of expression $$\frac{XA+XB}{XC+XD+XE}$$ is the lowest. I tried using Ptolemy's theorem but don't know how to make ...
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0answers
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Max function with absolute values for arbitrary number of arguments

The maximum between two numbers $x$ and $y$ can be easily written as $$ max(x,y) = \frac12\left(x+y +|x-y|\right). $$ We can obviously generalize this to any number of arguments as $$ max(x_1,\dots,...
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3answers
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If $x$ and $y$ are acute, and $\sin y = 3 \cos (x+y) \sin x$⁡, then find the maximum value of $\tan y$

Given $x,y$ are acute angles such that $$\sin y = 3 \cos(x+y)\sin x$$ Find the maximum value of $\tan ⁡y$. Attempt: We have $$\begin{aligned} 3(\cos x \cos y - \sin x \sin y) \sin x & = \sin ...
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How to maximize the fraction of the square of the sum of some sine and cosine functions

I have a problem with maximizing the fraction of the square of the sum of some sine and cosine functions. Denote $\mathbf{x} \triangleq {(x_i)}_{i \in \{1,2,...,N\}}$. The problem is shown below. $$\...
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question on max function with subscripts

i'de like to know, what this expression mean: $\max \limits_{o\le s\le t}\frac{K}{F(s)}$ 1) search for all values of $F(s)$ among $0\le s\le t$ and takes the maximum value or 2) search for all the ...
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1answer
67 views

Maximum value of function $f(x)=\frac{x^4-x^2}{x^6+2x^3-1}$ when $x >1$

What is the maximum value of the $$f(x)=\frac{x^4-x^2}{x^6+2x^3-1}$$ where $x > 1$ . My try Unable to solve further.
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1answer
33 views

Optimizing over an Integral

I have to solve the following optimization problem: $max_\tau \int_\underline{\epsilon}^\bar{\epsilon} \tau(1-\tau)^\epsilon d\epsilon$ How does one solve these with the integral. I think I may be ...
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1answer
33 views

Second derivative Test becomes zero

Consider the domain $D = \{ (𝑥,𝑦) ∈ ℝ^2:𝑥 ≤ 𝑦 \}$ and the function $ℎ: 𝐷 → ℝ$ defined by $ℎ((𝑥,𝑦)) = (𝑥 −2)^4 +(𝑦−1)^4$, $(𝑥,𝑦) ∈ 𝐷$. Find the minimum value of $h$ in the domain $D$: a) $...
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0answers
31 views

Finding extrema of a tri-variable function under a constraint

We need to find extremum of $$f(x,y,z) = yz$$ under the constraint $$g(x,y,z) = 2x^2 + 3y^2 + z^2 - 12xy + 14xz - 35$$ Using the technique of Lagrange Multipliers, leads to four simultaneous ...
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4answers
61 views

What is the minimum value of $|x| + |2x+1|+|3x+2|+\cdots+|99x+98|$?

What is the minimum value of the following? $$A = |x| + |2x+1|+|3x+2|+\cdots+|99x+98|$$ What I've tried so far: Since $|x| = |-x| $ it is clear that $|3x + 2|$ = $|-3x - 2|$, $|5x + 4| = |-5x-4|$ ...
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3answers
43 views

Maximize a bivariate function under constraints by Lagrange multipliers

I have been given a function $f(x,y)= x^2 + y^2$ whose maximum and minimum values have been sought (if existent) under the constraint that $3x^2 + 4xy +16y^2=140$. This looks pretty much to be a ...
2
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6answers
79 views

Find shortest distance from the parabola $y=x^2-9$ to the origin.

Find shortest distance from the parabola $y=x^2-9$ to the origin. First, I find minima of $\sqrt{x^2+(x^2-9)^2}$, so use derivative and ... Is have an easier way?
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4answers
198 views

Finding maxima of a function $f(x) = \sqrt{x} - 2x^2$ without calculus

My question is how to prove that $f(x) = \sqrt x - 2x^2$ has its maximum at point $x_0 = \frac{1}{4}$ It is easy to do that by finding its derivative and setting it to be zero (this is how I got $x_0 ...
4
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2answers
48 views

Finding extrema of function

We have the function $$f_a(x)=\frac{a-x}{\ln (a-x)}, \ a\in \mathbb{R}$$ To get the domain of that function we have to consider the following restrictions: $$\begin{cases}\ln (a-x)\neq 0 \\ a-x>0 ...