Questions tagged [cauchy-schwarz-inequality]

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

638 questions
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Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
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Prove that $3 \le \sum_{cyc}a\sqrt{b^3 + 1} \le \sum_{cyc}ab^2 + 3$ where $a, b, c \ge 0$ and $a + b + c = 3$.

$a$, $b$ and $c$ are non-negatives such that $a + b + c = 3$. Prove that $$\large 3 \le a\sqrt{b^3 + 1} + b\sqrt{c^3 + 1} + c\sqrt{a^3 + 1} \le \frac{ab^2 + bc^2 + ca^2}{2} + 3$$ This problem is ...
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Let $x$, $y$, $z$ be positive numbers. Prove that $\sqrt{x(3x+y)}\ + \sqrt{y(3y+z)}\ + \sqrt{z(3z+x)}\ \leq\ 2(x+y+z)$

Let $x$, $y$, $z$ be positive numbers. Prove that $\sqrt{x(3x+y)}\ + \sqrt{y(3y+z)}\ + \sqrt{z(3z+x)}\ \leq\ 2(x+y+z)$ Professor says Cauchy-Schwarz theory should be used.
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Maximum and minimum of $\cos^2x+\sin^2y$, where $x-y=\pi/4$ and $0\leq x\leq \pi$

In the book "Calculus of several variables" by Sege Lang in page 144 the author proposes the following problema: Find the extreme values of the function $$f(x,y)=\cos^2x + \cos^2y$$ subject to the ...
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$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$ given that where $x, y, z > 0$ and $xyz = \frac{1}{2}$.

$x$, $y$ and $z$ are positives such that $xyz = \dfrac{1}{2}$. Prove that $$\frac{xy}{z^2(x + y)} + \frac{yz}{x^2(y + z)} + \frac{zx}{y^2(z + x)} \ge xy + yz + zx$$ Before you complain, this problem ...
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Calculate the maximum value of $\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$ where $3a + 4b + 5c = 12$

$a$, $b$ and $c$ are positives such that $3a + 4b + 5c = 12$. Calculate the maximum value of $$\frac{ab}{ab + a + b} + \frac{2ca}{ca + c + a} + \frac{3bc}{bc + b + c}$$ I want to know if there are ...
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Let n be a positive integer such that

$\sin {\frac {π}{2n}}+\cos {\frac {π}{2n}}=\frac{\sqrt n}{2}$. Then A) $6\le n\le8$, B) $4\lt n\le8$, C) $4\le n\le 8$, D) $4 \lt n\lt8$, I couldn't get started in solving it. That's why ...
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Minimize this real function on $\mathbb{R}^{2}$ without calculus?

When it comes to minimizing a differentiable real function, calculus comes into play immediately. If $f: (x,y) \mapsto (x+y-1)^{2} + (x+2y-3)^{2} + (x+3y-6)^{2}$ on $\mathbb{R}^{2}$, and if one is ...
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Find Maxima and Minima of $f( \theta) = a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta$
Show that, whatever the value of $\theta$, the expression $$a \sin^2 \theta + b \sin \theta \cos \theta + c \cos^2 \theta\$$ Lies between $$\dfrac{a+c}{2} \pm \dfrac 12\sqrt{ b^2 + (a-c)^2}$$ My ...