Questions tagged [cauchy-schwarz-inequality]

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

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Maximizing with Cauchy-Schwarz inequality

I want maximize the function $f(x)=\cos(x)+\sin(x)\cdot\cos(x)$, with $x \in (0,\frac{\pi}{2})$. By derivation $f'(x)=0 \Rightarrow x=\frac{\pi}{6}$. But, if we write $f(x)=\cos(x)+\frac{1}{2}\sin(2x)$...
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Knowing $x,y,z\ge0$ prove $x^2+xy^2+xyz^2\ge4xyz-4$

Knowing $x,y,z\ge0$ prove $x^2+xy^2+xyz^2\ge4xyz-4$ I thought that I should rearrange this inequality to be somewhat of the form of Schur's Inequality and WLOG I assumed $x\ge y\ge z$. Trying this way ...
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The Promotion of Cauchy-Schwarz Inequality in Probability Theory

In probability theory, if we have $\xi$ and $\eta$ as random variables, the Cauchy Inequality can be designated as: $$E(\xi \eta) \leqslant \sqrt{E(\xi^2)E(\eta^2)}$$ Then I have an idea to make an ...
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Finding the value range using Cauchy-Schwarz inequality.

English is not my mother tongue so my English may sound weird. When real numbers $x,y$ satisfies $\sin x+3 \sin y=1$,let $z=\cos x+3 \cos y$. Find the range of $z$. I came across to this problem when ...
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Cauchy-Schwarz Inequality Proof in $\mathbb{R}^n$

Let $n \in \mathbb{N}$ and let $\sum_{i=1}^n x_i = 1$ where $x_i$ are positive real numbers. Use the Cauchy-Schwarz inequality to show that $$\sum_{i=1}^n \frac{1}{x_i} \geq n^2$$ The Cauchy-Schwarz ...
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Equivalent to Cauchy-Schwarz inequality

This is a question form a sample paper whose solutions where not published: Let $a$ and $b$ be two nonzero vectors in $3$ dimensions. The orthogonal projection of $b$ onto $a$ is$$c=(e\cdot b)e,$$...
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Equality characterization of the reverse Cauchy-Schwarz inequality in a Lorentzian manifold

Let $(M, g)$ be a Lorentzian manifold (signature -++...) with a time orientation and suppose that $v, w \in T_pM$ are causal vectors that are in the same light cone(ie, both future-directed or past-...
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The proof said to use the quadratic polynomial: $$P(t) = \sum_{i=1}^n(a_it - b_i)^2$$ by which we notice that $P(t) \ge 0$ , but then this is the part which I didn't understand where we conclude that ...
Suppose $a_i>0, b_i>0, c_i>0 \; \forall i = 1, 2, \dots, n$, and $$\sum_{i} a_i b_i \ln(\frac{c_i}{b_i}) \geq 0,$$ where $a_i, b_i, c_i$ are not constant over $i$. Moreover, \sum_{i}a_ib_i\...