# Tagged Questions

Questions on proving, manipulating and applying inequalities.

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### Proofs involving positive real numbers

I have two questions related to positive real numbers: If a and b are two vectors of positive random integers (no specific statistical distribution) and size N by 1 , we want to prove that the inner ...
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### Find maximum without calculus

Let $f:(0,1]\rightarrow\mathbb{R}$ with $f(x)=2x(1+\sqrt{1-x^2})$. Is it possible to find the maximum of this function without calculus? Possibility through some series of inequalities?
Consider the system: \begin{cases} \dfrac{dx}{dt} = y \\[12pt] \dfrac{dy}{dt} = -(1+x^{2})\,y-\sin(x) \end{cases} $(0,0)$ is a critical point of this system and I need to show that it is ...
Assume $A,B \in M_n(\Bbb{R})$ are positive definite matrices, show that $$\text{Tr}(AB)\leq \text{Tr}(A)\text{Tr}(B)$$ I only prove it for $n=2$, it is straightforward calculate.but when $n \geq ... 2answers 230 views ### Find Minimum value of$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$Given:$x,y\in (-\sqrt2;\sqrt2)$and$x^4+y^4+4=\dfrac{6}{xy}$Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ? 1answer 43 views ### Rational number inequality proof Show that if$x > 1$is a real number and if$a < b$are rational numbers, then$0\le x^a \le x^b$. My professor told me that I'm supposed to use some$x^c$, such that$c\epsilonQ$>$0$. ... 1answer 39 views ### inequality for real-valued Gaussian sums I saw the following Lemma in an article: Let$\mathbf{b}\in \mathbb{R}^N$be fixed, and let$\mathbf{\epsilon}\in \mathbb{R}^N$be a random vector whose N entries are i.i.d. random variables drawn ... 1answer 62 views ### Prove:$ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $Let$a,b,c>0$such that$a+b+c=1$Prove:$ \sum\frac{ab}{a^2+b^2}+\frac{1}{4}(\sum\frac{1}{a})\geq\frac{15}{4} $I don't have any idea. You guy have any idea?? 2answers 18 views ### Variable intervals from system of inequalities I have this system of inequalities: and I need to find possible intervals of i and j. Looking at the graph output from ... 0answers 109 views ### Prove that$\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}\$
Prove that $$\left\vert\prod_{k=1}^{n}{\sin (k)}\right\vert\leq\prod_{k=1}^{n-1}{\sin \left(\frac{k\pi}{n}\right)}\quad\forall n\in\mathbb{N}\backslash\{1\}.$$ Please show all passages and what ...