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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).

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).

As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.

A real-valued function $f$ defined on a domain $X$ has a global (or absolute) maximum point at $x^∗$ if $f(x^∗) \ge f(x)$ for all $x$ in $X$. Similarly, the function has a global (or absolute) minimum point at $x^∗$ if $f(x^∗) \le f(x)$ for all $x$ in $X$. The value of the function at a maximum point is called the maximum value of the function and the value of the function at a minimum point is called the minimum value of the function.

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Find the minimum value of $\sqrt{(x_1-x_2)^2 + (2-x_1-x_2)^2 + (2x_1-3x_2)^2}$

This is a question from a practice workbook for a college entrance exam. Find the minimum value of - $$\sqrt{(x_1-x_2)^2 + (2-x_1-x_2)^2 + (2x_1-3x_2)^2} \text{ where } x_1,x_2 \in R$$ I think it will possibly involve the decomposition of this…
First Year in India
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finding the max of a function

This question popped up during a nights drinking and it has been bugging me ever since. the conditions are: $$\left\{\begin{matrix} x+14y\leq 1820\\ x+25y\leq 2162.5\\ x\geq 0\\ y\geq 0 \end{matrix}\right.$$ The question is what is the maximum…
attle
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Maximize function consisting of Sum, Vectors and Signum function

Edit: Thanks for edit :-) Can I ask please, how to maximize $$\sum ({v_3 \text{sign}({xv_1+yv_2)}})\quad\text{where}$$ (edit: sum of i in 1..1000 ... 1 to length(v1) .. 3 vectors of same length $$\sum_{i=1}^n ({v_3[i]…
user2120
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Find the extrema of $f(x)=(x^2+2x-7)e^{|x+2|}$

I need to find the extrema of $f(x)=(x^2+2x-7)e^{|x+2|}$. The first derivative is $e^{|x+2|}(2x+2+(x^2+2x-7)(|x+2|)')$. Looking at two cases where $x<-2$ or $x>-2$ gives two extrema - at the points -3 and 1. And now comes the real question: Is there…
Kotaka Danski
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Proof for $\max(A+B) = \max(A) + \max(B)$ correct?

Hi I'm trying to prove the following: Let $A, B \subset \mathbb {Q}$ and $A+B = \{a+b: a\in A, b \in B\}$. Suppose $\max(A)$ and $\max(B)$ exist, show that $\max(A+B)$ also exists and that $$\max(A+B)= \max(A) + \max(B)$$ I have the following proof…
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Max of difference and difference of max

Considerva function $f(x)$ and a function g(x) and suppose they both have a maximum. They may take positive or negative values. Is it true that $$ \max_x (g(x)-f(x))\leq max_x g(x)-\max_x f(x) $$ Or the way around holds? Or we can't say?
Star
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Difference of maximum functions over finite alternatives

I wish to find an upper bound on $c$, where $c$ is defined as $$c = \max\{a_1,a_2,a_3\} - \max\{b_1,b_2,b_3\}$$ Is it true that $c\le\max\{a_1-b_1,a_2-b_2,a_3-b_3\}$?
jonem
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how to find the maxima of $\sin^3x+\cos^3x$ without using derivative

With derivative ,it's easy to do. $$3 \sin ^2(x) \cos (x)-3 \sin (x) \cos ^2(x)=0$$ $x=0,\max=1$ but without it,how?
AsukaMinato
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Maximum and minimum value of $\theta(x)=\int_0^x\frac{\sin z}{z}dz$

Let $\theta(x)=\int_0^x\frac{\sin z}{z}dz$, x>0. Then $\theta(x)$ has (A) maximum for $x = nπ$, n = 2, 4, 6, . . . . . . . (B) minimum for $x = nπ$, n = 1, 3, 5, . . . . . . . (C) maximum for $x = nπ$, n = 1, 3, 5, . . . . . . . (D) minimum for…
Samar Imam Zaidi
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$xy + ax^2 + bx + c = 0 , x+y$ has relative max or min if...

Options : $b^2-4ac > 0$ $b^2/4ac >0$ $b/(c-1) >0$ $c/(a-1)>0$ $a/(b-1)>0$ Relative max or min is local max or min, means max or min at certain open interval.. $xy + ax^2 + bx + c = 0 $ $ax^2 + (y+b)x + c = 0 $ if its a qudratic fuction. Then the…
Dini
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factor: $x^{\alpha-2}~(1-x)^{\beta-2}~\bigg((\alpha-1)(1-x)-(\beta-1)x\bigg)=0$

I started with a proof to find the maxima of a beta distribution $Beta(x: \alpha, \beta)$. This involves finding the derivative of the beta distribution and equating it with zero to yields the following root…
Leaky Capacitor
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Find the min value of $\frac{1}{x+\frac{1}{y+\frac{1}{z}}}$, if $x\ne y \ne z$ and $x,y,z\in {1,2,3,4,5}$

My answer is $\frac{5}{29}$, I just use logic to substitute numbers in the expression, but I can't prove my answer If this expression be minimum then the denominator should be the greatest, so I just let x=5 and I need to let…
nar
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Ask about global maximum and global minimum?

The temperature distribution in a metal rod given by the following function of the position $x \in \mathbb{R}$: $$T(x) = \frac{1 + 2x}{2 + x^2}$$ What is the maximal and minimal temperature in the metal rod? $T'(x) = 0$ when $x = 1$ or $x = -2$. But…
Mike LoongBoong
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Global Maxima of $x^2$ (can infinity be the maxima?)

Wolfram defines Global Maxima as : A global maximum, also known as an absolute maximum, the largest overall value of a set, function, etc., over its entire range. As per the definition, I'm not sure if $\infty$ can be considered as the global…
Mathejunior
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Boat and Arc Length problem

Boat men wants to reach at exact opposite point of river,which is 40m wide. Speed of boat in still water is 5m/sec and that of river is 2 m/sec . Find following 1- Angle at which he needs to start journey from starting point. 2-length of arc…
Abhishek Vidyarthi
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