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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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Assume that $1a_1^2+2a_2^2+\cdots+na_n^2 = 1,$ where the $a_j$ are real numbers. As a function of $n$, what is the maximum value of...

Assume that $1a_1^2+2a_2^2+\cdots+na_n^2 = 1,$ where the $a_j$ are real numbers. As a function of $n$, what is the maximum value of $(1a_1+2a_2+\cdots+na_n)^2?$ Thanks for all your help!
Quantum Pizza
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Assume that $ 1a_1+2a_2+\cdots+20a_{20}=1, $ where the $a_j$ are real numbers and that these values minimize...

Assume that $\ 1a_1+2a_2+\cdots+20a_{20}=1, $ where the $a_j$ are real numbers and that these values minimize $[1a_1^2+2a_2^2+\cdots+20a_{20}^2.]$ Find $a_{12}$. Any help would be greatly appreciated. Thanks!
Quantum Pizza
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global minimum for $2x^3+3x^2-12x+1$ on intervals between 0 and 2

How would I solve this? The maximum was straight forward, x = 1. But I can't find the minimum within the interval using derivatives. Any help would be appreciated. Thanks in advance.
21rw
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Let $x$ and $y$ be real numbers satisfying $\frac{x^2y^2 - 1}{2y-1}=3x.$...

Let $x$ and $y$ be real numbers satisfying $\frac{x^2y^2 - 1}{2y-1}=3x.$ Find the largest possible value of $x.$ I'm not sure how to do this question. Any help is greatly appreciated!
Quantum Pizza
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How to find global min and max of $2x^2+\cos(2x)$

It is bounded between $\pi/2$ and $2\pi$. Is the mean value theroem involved in solving this? Any help will be appreciated. Thanks in advance.
21rw
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Why do you draw the triangle as in the picture below and not in any other way?

In the closed area limited by the graph $y = 4-x^2$, an isosceles triangle is inscribed. The triangle has its top angle in the origin and its base is parallel to the x-axis. Decide the triangle's maximal area. My question is the following: Why do…
Andreas
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Maxima/Minima given a constraint

Let $f$ and $g$ be functions on $R^2$ defined respectively by $$f(x,y)=\frac{1}{3}x^3+\frac{3}{2}y^2+2x$$ and $$g(x,y)=x−y$$ Consider the problems of maximizing and minimizing $f$ on the constraint set $$C=\{{(x,y)∈R^2:g(x,y)=0\}}$$ (a) $f$ has a…
nsus
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What's the condition to exist a real global maxima or minima of a function $f:\mathbb{R}\mapsto \mathbb{R}$?

What's the condition to exist a real global maxima or minima of a function $f:\mathbb{R}\mapsto \mathbb{R}$? At first I thought that to test if this is true I could take the image of any of its derivatives, and if any of them is a proper subset of…
Garmekain
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Extremum of a monotonic function combination

If solution of $f'(x) /g'(x)= \lambda $ is an extremum point of $y = f(x) - \lambda \,g(x) $ then can it be shown that $f(x)$ and $g(x)$ are monotone functions?
Narasimham
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Equations of condition

In this example(Calculus Made Easy pg 126): $\frac{4R^2-2x^2}{\sqrt{4R^2-x^2}}=0$ simplifies to $4R^2-2x^2=0$ and therefore $x=R\sqrt{2}$ Why does the denominator not come into play when we equate to zero?
Isosceles
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Local maximum and global maximum of $\sin$

In book it is written that $\sin(x)$ has both local maximum and global maximum at $\pi/2$ but the highest value $\sin$ can have is $1$ and that is at $\pi/2$. Should not it be only global maximum?
Khan Saab
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Minimise sum of increasing functions with linear constraint

Let $f(x)$ and $g(x)$ be two increasing continuous functions. Given that $x_1 + x_2 = k$, show that the minimum of $f(x_1) + g(x_2)$ occurs where $f(x_1) = g(x_2)$
Trent Gm
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The $\min(\cdots ,\cdots)$ function

I have a question (might turn out silly, but I am missing something!) based on the minimum function. Does $|x-y|+|y-z|\ge |x-z| \implies \min(|x-y|,1)+\min(|y-z|,1)\ge \min (|x-z|,1)\ \ \ ?$ $x,y,z\in \Bbb{R}$
Qwerty
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The correct statements are

For every pair of continuous functions $f, g$ $: \left[0, 1\right] \rightarrow R $, such that $max(f(x) : x \in \left[0, 1\right]) $$= $ $max( g(x) : x \in \left[0, 1\right] )$ , the correct statement(s) is(are) (1)$(f(c))^2 + 3f(c) = (g(c))^2 …
Aakash Kumar
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Find minimum value of the algebraic expression

With a,b,c>0 and a+b+c=3 find the minimum value of $$P = \frac{a}{b^2+1} + \frac{b}{c^2+1} + \frac{c}{a^2+1}$$ This problem is the last part of my second midterm exam which i couldn't do since it is the hardest one. For me, i think that the minimum…
ruh roh
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