Análise Real : Números Reais (desigualdade de Cauchy-Schwar
Enviado: Dom Fev 22, 2015 18:00Use a desigualdade de Cauchy-Schwarz para mostrar que:
a) Se
, então ![\frac{1}{2}(\sqrt[2]{a}+\sqrt[2]{b}+\sqrt[2]{c}+\sqrt[2]{d})\geq\sqrt[2]{a+b+c+d}](/latexrender/pictures/3a488cde3ffed1afbc446372ab9b2d84.png "\frac{1}{2}(\sqrt[2]{a}+\sqrt[2]{b}+\sqrt[2]{c}+\sqrt[2]{d})\geq\sqrt[2]{a+b+c+d}")
;
b)![cos\theta.sen\varphi+sin\theta.sin\varphi+cos\varphi\leq\sqrt[]{3}](/latexrender/pictures/7f205316b288935743858e178bf84ac3.png "cos\theta.sen\varphi+sin\theta.sin\varphi+cos\varphi\leq\sqrt[]{3}")
, para quaisquer 
;
c) Se
e 
, então 
a) Se
, então ;b)
, para quaisquer ;c) Se
e , então ![{(\sqrt[]{a}-\sqrt[]{b}})^{2}\geq 0\Rightarrow a-2\sqrt[]{a}\sqrt[]{b}+b\succeq 0](/latexrender/pictures/86a4e2b148e79ab349d6649bbc3ed1d1.png "{(\sqrt[]{a}-\sqrt[]{b}})^{2}\geq 0\Rightarrow a-2\sqrt[]{a}\sqrt[]{b}+b\succeq 0")
![\Rightarrow a+b\succeq 2\sqrt[]{a}\sqrt[]{b}\Rightarrow a+2\sqrt[]{a}.\sqrt[]{b}+b\succeq 4\sqrt[]{a}\sqrt[]{b}\Rightarrow {(\sqrt[]{a}+\sqrt[]{b}})^{2}\succeq 4\sqrt[]{a}\sqrt[]{b}](/latexrender/pictures/03aa48a976376a823cdd961341f97555.png "\Rightarrow a+b\succeq 2\sqrt[]{a}\sqrt[]{b}\Rightarrow a+2\sqrt[]{a}.\sqrt[]{b}+b\succeq 4\sqrt[]{a}\sqrt[]{b}\Rightarrow {(\sqrt[]{a}+\sqrt[]{b}})^{2}\succeq 4\sqrt[]{a}\sqrt[]{b}")
![\Rightarrow 1/2(\sqrt[]{a}+\sqrt[]{b})\succeq \sqrt[]{a.b}\succeq \sqrt[]{a+b}](/latexrender/pictures/70aa8984c260215fc8533e5a2ae01f10.png "\Rightarrow 1/2(\sqrt[]{a}+\sqrt[]{b})\succeq \sqrt[]{a.b}\succeq \sqrt[]{a+b}")
...
![\preceq \left|sen\thet.cos\thetasen\varphi \right|+\left|sen\theta.sen\thetacos\varphi \right|+\left|cos\varphi \right|\preceq (\sqrt[]{3}/2).(\sqrt[]{3}/2)+(\sqrt[]{3}/2).(\sqrt[]{3}/2)+(\sqrt[]{3}/2)](/latexrender/pictures/ae0f8d368ab05c95b0c8561d2dcf4a5e.png "\preceq \left|sen\thet.cos\thetasen\varphi \right|+\left|sen\theta.sen\thetacos\varphi \right|+\left|cos\varphi \right|\preceq (\sqrt[]{3}/2).(\sqrt[]{3}/2)+(\sqrt[]{3}/2).(\sqrt[]{3}/2)+(\sqrt[]{3}/2)")
=![2/3+(\sqrt[]{3}/2)\prec (\sqrt[]{3}/2).(\sqrt[]{3}/2)=\sqrt[]{3}](/latexrender/pictures/95fde11370746158fb213481d132c49d.png "2/3+(\sqrt[]{3}/2)\prec (\sqrt[]{3}/2).(\sqrt[]{3}/2)=\sqrt[]{3}")
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