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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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increasing and decreasing intervals of a function

$f(x)= x^3-4x^2+2$, which of the following statements are true: (1) Increasing in $(-\infty, 0)$, decreasing in $(\frac{8}{3}, +\infty)$ (2) Increasing in both $(-\infty, 0)$, and $(\frac{8}{3}, +\infty)$ (3) decreasing in both $(-\infty, 0)$, and…
Myshkin
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Minoration of max function

I am wondering if we have the following minoration of the $\max$ function : $$ \forall a, b, c \in \mathbf{R} ~~~~~ \max(a, b, c) \geq \dfrac{1}{3} ( a+b+c) $$
Amine HANINI
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How can it be shown that $x^x = a$ , has no real solutions when $a < (1/e)^{(1/e)}$

Using calculus to find the minima: $$y(x) = x^x$$ $$ln(y) = x*ln(x)$$ $$(1/y)*\frac{dy}{dx} = ln(x) + x*\left(\frac{1}{x}\right) = ln(x) + 1$$ $$\frac{dy}{dx} = y*(ln(x) + 1)$$ $$\frac{dy}{dx} = (x^x)*(ln(x) + 1)$$ Though arriving at this next step,…
Jack Freeth
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Max and Min value of a function on a circle

Find the maximum and minimum values of the function $f(x,y) = 5x^2 + 2xy + 5y^2$ on the circle $x^2 + y^2 = 1$. After substituting the equation of the circle in that of the function and then equating $f'(x) = 0$, I get the values of $y$ to be…
Rijo Paul
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maximum and minimum of $f(x)=\frac{x}{1+x^2}$

Is there a way how to calculate maximum and minumum of $f(x)=\frac{x}{1+x^2}$ without derivative?
abcabcabc
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Are bottom and top the synonyms of minimum and maximum of the partially ordered set respectively

Wikipedia states: The least and greatest element of the whole partially ordered set plays a special role and is also called bottom and top, or zero (0) and unit (1), or ⊥ and ⊤, respectively. If both exists, the poset is called a bounded poset.…
user715522
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Find points on curve $ax^2 + 2bxy + ay^2 = c$ whose distance from origin is minimal

Find the points on the curve $$ax^2 + 2bxy + ay^2 = c$$ where $c > b > a > 0$ and whose distance from the origin is minimum. My approach: $(x_1)^2+(y_1)^2=D^2$ Putting the values $ax_1^2 + 2bx_1y_1 + ay_1^2 = c$ $aD^2 + 2bx_1y_1 = c$ I am not able…
Samar Imam Zaidi
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Maximizing $\prod_{i=0}^n|x-\frac{i}{n}|$

Let $x_i=\frac{i}{n}\;, i=0,1,...,n\;$ be $\;n+1$ equidistant points on the interval $[0,1]$. Prove: The maximum value of: $$\prod_{i=0}^n|x-\frac{i}{n}|$$ in the interval $[0,1]$ is achieved for $x\in (0,\frac{1}{n})$ or $x\in (\frac{n-1}{n},1)$…
user401516
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find minimum and maximum value of a function with 2 variable in a given range

Given a function with $2$ variable $f(x, y)$ , I need a general rule to find max and min of this function in a range for $a\leq x\leq b, c\leq y\leq d$. I know how to differentiate two variable function but that would only give local maximum and…
Roushan Singh
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Find the maximal and minimal values of the function f(x,y)

Find the maximal and minimal values of the function f given by: $$f(x,y) = 2x^2+y^2-4y-4x+1$$ on the triangle bounded by the lines $x = 0,y = 2x, y = 2$ Here is my work: $$f_x =4x-4=0 => x=1$$ $$f_y =2x-4=0 => y=2$$ It appears that (1,2) is not…
0oRAKo00
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local/global extremum

Hi guys just wanted to clear up this notion of local/global extremum at an interior point. I would like to know what's the proper definition for a function having a local/global extremum. I know a local extremum can be a maximum or minimum value…
John
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Finding Maximum and Minimum values and Justifying if they exist

Find the maximum and minimum of each of the following subsets of R, if they exist justify the answer. 1) A = (0,3] 2) B = {x is a member of Z (integers): x < 1/2} 3) C = Q (Rationals) intersecting [-1, pi] 1) for the first question i would say that…
John
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I want to minimise a function with multiple variable

$y = \sqrt((x -16)^2 + (x - 6)^2) + \sqrt((x - q)^2 + (x- (-q +10))^2) + \sqrt((q + 7)^2 + ((-q+10) -13)^2)$ where ${x >= 10}$ ${x<=20}$ ${q>=-10}$ ${q <= 0}$ ${-q + 10 >= 10}$ ${-q +10 <= 20}$ I know how to minimise a function with 1 variable. I…
Jean-Claude Lemieux
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Finding minimum value of bloom filter function

Bloom filter definition from Wikipedia: A Bloom filter is a space-efficient probabilistic data structure, conceived by Burton Howard Bloom in 1970, that is used to test whether an element is a member of a set. False positive matches are…
Robur_131
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What does $\max_{i=1}^k (f_i(x) - q_i)$ mean? Is it a sum?

What does $\max_{i=1}^k (f_i(x) - q_i)$ mean? Is it a sum? This is the Chebyshev achievement function. And it's taken in "$L^{\infty}$ sense". The optimization problem related to this is written: $$\min_{x \in S}\max_{i=1}^k (f_i(x) - q_i) $$ Does…
mavavilj
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