# 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).

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### If $K\subset \mathbb{R}^2$ is closed, can we find a smooth function $f:U\to \mathbb{R}$, s.t. $U$ is open, $K\subset U$ and $f$ is "nowhere" constant?

In John Lee's Introduction to Smooth Manifolds, p. 47, we have the following result: Theorem 2.29 (Level Sets of Smooth Functions). Let $M$ be a smooth manifold. If $K$ is any closed subset of $M$, ...
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### Is it correct to follow argmax with an inequality? $\arg \max_{x}\{f(x)<g\}$

In some papers, I saw a formula like this：$\arg \max_{x}\{f(x)<10\}$. I guess the author here wants to express that when $f(x)<10$, the largest of all $x$. This is different from the definition ...
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### Prove maximum of $p\cdot(1-p)^{n-1}$ is at $p=\frac{1}{n}$ without differentiation

I would like to do a proof with my high school students that for $0\le p\le 1$, the maximum of $p\cdot(1-p)^{n-1}$ is reached for $p=\frac{1}{n}$. Proving this with differentiation is quite trivial, ...
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### solution-verification | What is the maximum number of distinct planes determined by three of the 25 points?

The problem Consider a plane $\alpha$, a line $d || \alpha$, five points $A,B,C,D,E$, any 3 non-collinear located on the $\alpha$ plane and the points $P_1,P_2,... , P_20$ distinct two by two, located ...
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### Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$

The problem Let $a,b\in \mathbb N$. If $a+b=101$ and $\sqrt{a}+\sqrt{b}$ has its maximum value then calculate $a\cdot b$ My idea $(\sqrt{a}+\sqrt{b})^2=a+b+ 2\cdot \sqrt{ab}= 101 + 2\cdot \sqrt{ab}$ ...
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### better method of solving quadratic / cubic

Problem : Let \begin{align} f(x) &= x^4 - 8x^3 + 18x^2 \\ g(x) &= 9x^2 - 64x\end{align} . Define $h : \mathbb{R}^+ \to \mathbb{R}$, $h(x)=f(x)-ag(x)$ for some real number $a$. If $h(x)$ ...
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### Calculate optimal spacing for magnetic field measurement using Gaussian Multivariate likelihood distribution

I posted this question on Physics exchange as well, but is rather mathematic :) I have a vertical magnetometer configuration, and measure lines on the ground. I want to calculate the optimal spacing, ...
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### Minimization of a function with exponential and 2nd degree polynomial

I would like to minimize the following function (from $\mathbb{R}$ to $\mathbb{R}$): $$f(x) = -2a(1+x)\exp(-\Vert xb-c \Vert^2) + (1+x)^2 d$$ where $a,d \in \mathbb{R}$ and $b,c \in \mathbb{R}^d$ So ...
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### Maximum value of $\displaystyle f(x)=\bigg|\sqrt{\sin^2(x)+2a^2}-\sqrt{2a^2-3-\cos^2(x)}\bigg|$

Maximum value of $\displaystyle f(x)=\bigg|\sqrt{\sin^2(x)+2a^2}-\sqrt{2a^2-3-\cos^2(x)}\bigg|$ What I try : Put $\displaystyle 2a^2-\cos^2(x)=t$ So we have $\displaystyle f(t)=\sqrt{1+t}-\sqrt{t-3}$...
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### Prove the area of a quadrilateral $A_1B_1C_1D_1$ has a local minimum at $A_1=A$
$ABCD$ is a cyclic quadrilateral. Its diagonals $AC,BD$ intersect at $P$. Let $E$ be the point on $AB$ such that $AE:EB=\tan\angle BAP:\tan\angle ABP$. Let $F$ be the point on $BC$ such that \$BF:FC=\...