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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.

3734 questions
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Minimize $\frac{x^2}{x-9} $ given that $x > 9$

Its given that $ x > 9 $ and I have to find minima of $$ y = \frac{x^2}{(x-9) } $$ I did this using three methods. Method 1). Let $$ f(x) = \frac{x^2}{(x-9) } $$ Using, calculus, we get $$ f'(x) = \frac{2x}{(x-9)} - \frac{x^2}{(x-9)^2} …
user9026
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Find the maximum value of $x_1x_2+x_2x_3+\cdots+x_nx_1$

Find the maximum value of $x_1x_2+x_2x_3+\cdots+x_nx_1$ if $x_1+x_2+\cdots+x_n=0$ and $x_1^2+x_2^2+...+x_n^2=1$. I can get by Cauchy-Schwarz that $x_1x_2+x_2x_3+\cdots+x_nx_1 \leqslant 1$, but can expression $x_1x_2+x_2x_3+\cdots+x_nx_1$ be smaller?
chaos
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Is there a name for functions that have multiple minima such that all minima are equal/one minima is less than another?

Is there a name for a concept like that I can search? I am completely okay if there is no name so far. For example, $f\left(x\right)=\sin\left(x\right)$ has infinite minima, but they are all at $f\left(x\right)=-1$. On the contrary,…
Jake's Lamp
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Can a function have equal maximum and minimum value?

When I heard that the maximum value can be smaller than the minimum value, it striked me whether they can both be equal. I want to enquire whether it is true.
Shruthi Sivakumar
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Find the minimum of function

Find the minimum of $f(x)=\sqrt{(1-x^2)^2+(2-x)^2}+\sqrt{x^4-3x^2+4}$. I haven't learned derivative so I tried to solve it with geometry.But in the end I failed. Maybe it can be seemed as the sum of the lengths of two line segments?
CBot
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Finding the maximum value of algebraic expression

What is the maximum value of $(ax+b)^\frac12 + (cx+d)^\frac12$ I did it using the concept $AM>=GM$ $\frac{(ax+b)^\frac12+(cx+d)^\frac12}2 \geq ((ax+b)^\frac12(cx+d)^\frac12)^\frac12$ And on solving this I found that $x\leq(b-d)/(c-a)$ And when…
Kshitij Singh
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Finding the least value of $y=3^{x-1}+3^{-x-1}$

If $y=3^{x-1}+3^{-x-1}$ where $x$ is real, then what is the least value of $y$? What I did: $$y=3^{x-1}+3^{-x-1}=3^x3^{-1}+3^{-x}3^{-1}$$ Let $3^x=z$, then $$y=\frac z3+\frac1{3z}$$ $$3y=z+\frac1z$$ Now what to do next? Is there any better way to…
mathematics
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Minimize $A=\frac{1+2^{x+y}}{1+4^x}+\frac{1+2^{x+y}}{1+4^y}$

For $a,b>0$. Minimize $$A=\frac{1+2^{x+y}}{1+4^x}+\frac{1+2^{x+y}}{1+4^y}$$ i think we let $2^x=a;2^y=b$ Hence $A=\frac{1+ab}{1+a^2}+\frac{1+ab}{1+b^2}$ We need pro $A\geq 2$(Wolfram Alpha) but $x,y$ is a very odd number and i can't find how to…
Word Shallow
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Maximum and Minimum of $\displaystyle \frac{36}{a}+\frac{25}{b}$

If $\displaystyle \frac{a^2}{36}-\frac{b^2}{25}=1$.Then range of $\displaystyle \frac{36}{a}+\frac{25}{b}$ is Using Trigonometric Substution, We have $\displaystyle a=6\sec\theta$ and $y=5\tan\theta$. Then we have expression $\displaystyle…
jacky
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Maximum value of an expression by equation

Let $a,b \in \mathbb{R}$ Given, $$a+b = a^2+b^2 $$ then what is the maximum value of $(a + 7b)$ ? No idea how to deal with this.
NewBeginner
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how to find maximum value without differentiating

let x be positive real number, find max possible value of the expression $$y = \frac{x^2 + 2 - \sqrt{x^4 + 4}}{x}$$ it can be found by differentiating, but is there no other way of finding it, like using AM $\geq$ GM. or any other method. i…
hemant kumar
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Does $-\min(a,b) = \max(-a,-b)$?

Disclaimer: I'm not a mathematician. I'm a developer (with physics degree). By trial and error I've found that: $$-\min(a,b)=\max(-a,-b)$$ $$-\max(a,b)=\min(-a,-b)$$ Is it true? Is there a proof of it so I can read it? (Ideally explained so simple…
SeeR
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Solve the given word problem based on maximizing functions.

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of the removed squares is…
Aditya
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Consider a Quadrilateral PQRS as given in the diagram with paths AB,BC and CD as shown. Then what is the minimum value of $(AB+BC+CD)^2 -20$?

I tried using coordinate geometry to find the distance AB,BC and CD and then finding their minimum by partially differentiating the equations. This yielded a rather complicated equation which I could not solve. I'm sure there must be a more direct…
Maven
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How to determine the minimum value of a 4th order polynomial without using calculus?

I have got a $4^{th}$ order polynomial, $$x^4+100x^2−2000x+10^4=f(x)$$ whose minimum value I need without using calculus. Is there any way except graph plotting?
Elin Das
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