# Questions tagged [cauchy-schwarz-inequality]

Problems with using C-S (Cauchy-Schwarz inequality)

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### Prove a vector integral inequality

Given continuous $f:\mathbb{R}\rightarrow\mathbb{R}^n$, $x_1<x_2$, how to prove $||\int_{x_1}^{x_2}f(x)dx|| \leq \int_{x_1}^{x_2}||f(x)||dx$? I think it needs to use Cauchy-Schwarz inequality, and ...
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### Find the Maximum value of $\frac{x}{\sqrt{x+y}}+\frac{y}{\sqrt{y+z}}+\frac{z}{\sqrt{z+x}}$

if $x$, $y$ and $z$ are positive real numbers such that $x+y+z=4$ Find the maximum value of $$S=\frac{x}{\sqrt{x+y}}+\frac{y}{\sqrt{y+z}}+\frac{z}{\sqrt{z+x}}$$ I tried as follows. The given ...
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### Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds.

Knowing that for any set of real numbers $x,y,z$, such that $x+y+z = 1$ the inequality $x^2+y^2+z^2 \ge \frac{1}{3}$ holds. I spent a lot of time trying to solve this and, having consulted some books,...
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### Showing that $(x_2 - x_1)^2 + (y_2 - y_1)^2 \leq 8$, where $x^2 + y^2 \leq 1$

Let $D^2$ be the unit disk in $\mathbb{R}^2$ and $(x_1, y_1), (x_2, y_2) \in D^2$. I can't show that $$(x_2 - x_1)^2 + (y_2 - y_1)^2 \leq 8.$$ Can someone give me a hint?