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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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A very general method for proving inequalities. Too good to be true?

Update I 'repaired' this method, but it changed a lot and I have some different questions, so I posted it separately here. As training for the olympiad, I have to solve a lot of inequalities. Recently, I found a very general method to solve…
Mastrem
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Finding the largest equilateral triangle inside a given triangle

My wife came up with the following problem while we were making some decorations for our baby: given a triangle, what is the largest equilateral triangle that can be inscribed in it? (In other words: given a triangular piece of cardboard, what is…
ShreevatsaR
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probability density of the maximum of samples from a uniform distribution

Suppose $$X_1, X_2, \dots, X_n\sim Unif(0, \theta), iid$$ and suppose $$\hat\theta = \max\{X_1, X_2, \dots, X_n\}$$ How would I find the probability density of $\hat\theta$?
Enzo
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Prove that $(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$

Prove that $f(x)=(1+x)^\frac{1}{x}+(1+\frac{1}{x})^x \leq 4$ for all $x>0.$ We have $f(x)=f(\frac{1}{x}), f'(x)=-\frac{1}{x^2}f'(\frac{1}{x}),$ so we only need to prove $f'(x)＞0$ for $0 < x < 1.$
lsr314
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What is the shape of the perfect coffee cup for heat retention assuming coffee is being drunk at a constant rate?

Find the optimal shape of a coffee cup for heat retention. Assuming A constant coffee flow rate out of the cup. All surfaces radiate heat equally, i.e. liquid surface, bottom of cup and sides of cup. The coffee is drunk quickly enough that the…
Michael McLaughlin
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Are these continued fractions of integrals known?

Crossposted on MathOverflow Motivated by Bowman and McLaughlin (2000) on polynomial continued fractions, I considered an extension to functional continued fractions, where the numerator of each fraction is the integral of the previous…
TheSimpliFire
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Show that the maximum value of this nested radical is $\phi-1$

I was experimenting on Desmos (as usual), in particular infinite recursions and series. Here is one that was of interest: What is the maximum value of $$F_\infty=\sqrt{\frac{x}{x+\sqrt{\dfrac{x^2}{x-\sqrt{\dfrac{x^3}{x+ \sqrt{…
TheSimpliFire
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Is the function $f(x) = \sum_{k=1}^\infty (-1)^{k+1} \sin(x/k)$ bounded on $\mathbb{R}$?

Consider the function $f: \mathbb{R} \to \mathbb{R}$ defined by the series: $$f(x) = \sum_{k=1}^\infty (-1)^{k+1} \sin(x/k)$$ For any fixed $x \in \mathbb{R}$, the series converges by the Alternating Series Test, so the function $f(x)$ is…
Malo
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What are the common abbreviation for minimum in equations?

I'm searching for some symbol representing minimum that is commonly used in math equations.
Darqer
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What is $\lim\limits_{n\to\infty}\frac1n \left(\text{maximum value of }\sum\limits_{k=1}^n\sin (kx)\right)$?

Consider $f(x)=\sum\limits_{k=1}^n\sin (kx), 0\le x \le \pi$. Here is the graph of $y=f(x)$ for $n=8$. I noticed that, as $n\to\infty$, the maximum value of $\frac1n f(x)$ seems to approach a limit of approximately $0.7246$. What is…
Dan
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Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides $a, b, c$ . The "square" which I will refer to, from now on, has all its four vertices on the sides of the triangle, and of course is completely…
Sawarnik
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Determine the maximum of $f(x) = x + \sqrt{4-x^2}$ without calculus

I was given this problem in a Calc BC course while we were still doing review, so using derivatives or any sort of calculus was generally forbidden. We were only doing review because a lot of people hadn't taken Calc AB due to scheduling issues so…
Copywright
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Does a polynomial that's bounded below have a global minimum?

Must a polynomial function $f \in \mathbb{R}[x_1, \ldots, x_n]$ that's lower bounded by some $\lambda \in \mathbb{R}$ have a global minimum over $\mathbb{R}^n$?
Charles Chen
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How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$?

Related question asked by me on MathOverflow: How to prove $e^x\left|\int_x^{x+1}\sin(e^t) \,\mathrm d t\right|\le 1.4$? This is a follow-up question to the question How to prove $ \mathrm{e}^x\left|\int_x^{x+1}\sin\mathrm e^t \mathrm d…
Maximilian Janisch
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Find $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$ on unit sphere

Let $a,b,c,d,e\in \mathbb{R}$ such that $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ or more precisely, find $$\max_{a^2+b^2+c^2+d^2+e^2=1} \min(|a-b|, |b-c|, |c-d|, |d-e|,…
math110
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