# 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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### Min-max principle for eigenvalues in 1d elliptic problem

I have the following eigenvalue problem: $$\begin{cases} -u''=\lambda u\\ u(0)=u(\pi)=0 \end{cases}$$ and I have to prove the following min-max priciple for eigenvalues: \...
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### Show that the absolute maximum of $f(x, y) = \frac{(ax+by+c)^2}{x^2+y^2+1}$ is $a^2 + b^2 + c^2$

I got this question on my exam today: show that the function $f(x, y) = \dfrac{(ax+by+c)^2}{x^2+y^2+1}$ has an absolute maximum whose value is $a^2 + b^2 + c^2$. I tried setting the gradient to the ...
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### Find the minimum of $f(x)=x^2-x+1+\sqrt{2x^4-18x^2+12x+68}$.

WA gives the result $9$. But how to solve it by applying inequalites?
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### Knowing that $f(x) = ax^4 + bx^3 + cx^2 + dx + e \ (a \ne 0)$ and $e > n$, how many extrema does the function $y = f'(f(x) - 2x)$ have?

Consider graph $f(x) = ax^4 + bx^3 + cx^2 + dx + e \ (a \ne 0)$ whose derivative's graph is illustrated as the following. Knowing that $e > n$, how many extrema does the function $y = f'(f(x) - 2x)$...
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### Find the parameters $a,b$ such that the distance of all the points of a subset of $\mathbb{R^2}$ to the line $y=ax+b$ is minimal

Let $A\subset \mathbb{R^2}$ be a non-empty, finite set. We define a function $f:\mathbb{R^2}\rightarrow\mathbb{R},$ $f(a,b)=\sum_{(x,y)\in A}||y-ax-b||^2$. Find the global minimum of this function. ...
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### Argmax of the product of positive functions [closed]

Let $f(x), g(x) \geq 0$. Then, I want to know if the following is true $$\arg \max_x [f(x)g(x)] = \arg\max_x f(x) \cdot \arg\max_x g(x)$$ And how one can prove it. I found a related question in this ...
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### Determine minimum values of a function

Find the $x$ values that minimize the function $f(x)$. $$f(x) = e^x + \frac{1}{m}(x-4)^2$$ I know I can determine the minimum values when the derivative of the function is $0$. So I have calculated ...
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### Continuity of a pointwise maximum function of probability distributions

Consider a pair of finite alphabets $\mathcal{X}$ and $\mathcal{Y}$. Let $P_{Y|X}$ be a conditional probability distribution and let $Q_Y$ be a full-rank probability distribution. I am looking at the ...
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### Minimize a function regarding two coupling variables

Given known matrics $A\in \mathbb R^{2\times 2}$ and known vectors $b\in \mathbb R^2, c\in \mathbb R^2$, for the two optimization variables $x\in \mathbb R^2$ and $y\in \mathbb R$, how to obtain the ...
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### Claim concerning the maximum and minimum of $\sin x \,\cos A+ \sin y\, \cos B$

I have this expression $$\sin x \,\cos A+ \sin y\, \cos B \, ,$$ with $x,y\in\mathbb{R}$ and $0\leq A,B\leq2 \pi$. Then based on $∣\cos A∣\leq 1$ and $∣\cos B∣\leq 1$, is this claim true for the ...
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### extremum points of function

I was wondering if I have a function F[x,t], (in my case a polynomial), and I find the extremum points, which are fractions, if the denominator of the extremum points vanishes, does this ensure ...
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### Maximize $\sum_k \frac{p_k}{\sum_{j \geq k} p_j}$ over the probability simplex?
Suppose that $p_1, \dots, p_n$ are nonnegative real numbers such that $p_1 + \cdots + p_n = 1$; denote the corresponding set of vectors by $\Delta_n$. I am interested in the following function, \$f \...