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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 maximum value of $\frac{101^{k/2}}{k!}$ where $k$ is a positive integer

The positive integer $k$ for which $\dfrac{101^{k/2}}{k!}$ is maximum. A) $9$ B) $10$ C) $11$ D) $101$ I have no idea how to solve this problem. At first, I thought of taking this a function and then finding the roots of its…
Piano Land
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Longest distance of a point from an ellipse

So here is a question I was trying to solve: The longest distance of the point $(a,0) $ from the curve $2x^2+y^2-2x=0$ is given by ? My Attempt: The curve in the question represents an ellipse. So I wrote the parametric form of the ellipse to get…
user185887
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For $x, y \in \mathbb{R}$, prove that $\max(x, y) = \frac{x + y + |x - y|}{2},$ and $\min(x, y) = \frac{x + y - |x - y|}{2}$.

Prove that for all real numbers $x$ and $y$, $$\max(x, y) = \dfrac{x + y + |x - y|}{2},$$ and $$\min(x, y) = \dfrac{x + y - |x - y|}{2}.$$ For any real number $x$, the absolute value of $x$, denoted $|x|$ is defined as…
Preston Garvey
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Is it possible to prove $\min\{\max\{a,b\},\max\{a,c\},\max\{b,c\}\}=\max\{\min\{a,b\},\min\{a,c\},\min\{b,c\}\}$ without using cyclic symmetry?

I want to prove the following identity: For every real numbers $a,b,c,$ \begin{gather*} \min\{\max\{a,b\},\max\{a,c\},\max\{b,c\}\}=\max\{\min\{a,b\},\min\{a,c\},\min\{b,c\}\}. \end{gather*} Clearly, let \begin{gather*} …
azc
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minimum surface are for a given volume

How to prove that for a given enclosed volume, a sphere has minimum surface area
user12345
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A problem on Minima. Proving sphere has minimum surface area for a given volume

Has the below question been answered here before? Prove: For a given enclosed volume, a sphere has minimum surface area. Please provide link or ways to solve it. I know it is a problem of Minima and involves finding derivative and second…
user12345
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For $a,b,c \in R$ and $a,b,c>0$. Minimize $A=a^3+b^3+c^3$

For $a,b,c \in R$ and $a,b,c>0$ satisfy $a^2+b^2+c^2=27$, minimize $$A=a^3+b^3+c^3$$
Word Shallow
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Is there something similar to Lagrange multipliers, but to get critical points of an $f$ on $g \leq m$ instead of on $g = m$?

I have $$f,g:\mathbb{R^n} \rightarrow \mathbb{R}, m \in \mathbb{R}$$ Under any conditions of 'goodness' of the functions, is there a way to get the critical points of $f$ on $D = \{ g \leq m \}$ (Suppose $g \geq 0$) Lagrange multipliers gives us a…
Guillermo Mosse
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Finding maximum and minimum with given constraints

$y=(10-a)(10-b)(10-c)$ $\text{where } (a+b+c)=3\text{ and each of (a,b,c) are positive}$ What is the maximum and minimum value $y$ can have? I have created this question with these small values for ease of understanding. From my trials in excel,…
Kiran
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The exact procedure of calculating $\max_{x} f(x,y)$?

What is the exact procedure to calculate $\max_{x} f(x,y)$? This is my opinion: Choose one certain $y = y_0$, and then scan all $x$,find the max value of $f(x, y_0)$ named $f_0$; Choose another certain $y = y_1$, and do the same thing, I can get…
onebook
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when $\underset{x\in I}{\max}|(x-x_0)(x-x_1)|$ is minimum

I'm trying to find out for what values of $x_0$ and $x_1$, $\underset{x\in I}{\max}|(x-x_0)(x-x_1)|$ becomes minimum for $I=[-1,1]$. Note that we only look at the function diagram in $[-1, 1]$. I looked at the diagram and guessed value of function…
user1253495
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Max/Min of linear function

Has anybody got an idea how the equations $\max\limits_{\partial B_r(x)}u = u\left(x+r \frac{Du(x)}{|Du(x)|}\right)$ $\min\limits_{\partial B_r(x)}u = u\left(x-r \frac{Du(x)}{|Du(x)|}\right)$ arise for linear and non-constant…
HelloEveryone
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Is there a general term that can be used for critical points not maximum or minimum?

Most references do not use or avoid using terms when discussing critical points that are not maximum or minimum. But for the sake of asking a general term that encompasses critical points which are not maximum or minimum, what would be the term? I…
AndroidV11
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Minima of a function without derivation

Let $a,b,c \in \mathbb{R}^+$. How can we find the minima of the following function: $$ f(x) = \sqrt{a^2+x^2} + \sqrt{(b-x)^2 + c^2} $$ without calculating the derivative? I showed that $a+c$ is a minorant, it is à minima iff $b=0$. But I couldn't…
odwimi
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The regular tetrahedron inscribed in the hemisphere.

Let there be four points in the space region $$x ^ 2 + y ^ 2 + z ^ 2 \leqslant1,z\geqslant 0,$$ and the distance between any two points is $d$, then find the maximum value of $d$ . I know it is equivalent to finding the length of the regular…
Mr.He
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