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Inegalități importante

 

1.  \(a>1\)      \({{a}^{k-1}}<{{a}^{k}}\) \(\forall \)\(k\ge 1\)

     \(a\in \left( 0,1 \right)\) \({{a}^{k}}<{{a}^{k-1}}\) \(\forall \)\(k\ge 1\)

2.  \(0<a\le b\) \(\Rightarrow \left( {{a}^{m}}-{{b}^{m}} \right)\left( {{a}^{n}}-{{b}^{n}} \right)\ge 0\) \(\forall \)\(m,n\in N\)

3.  \(a+\frac{1}{a}\ge 2\) \(\left( \forall  \right)\) \(a>0\)  \(a+\frac{1}{a}\le -2\) \(\forall \) \(a<0.\)

              

7.  \({{a}^{2}}+{{b}^{2}}+{{c}^{2}}\ge ab+bc+ca\[\forall \]a,b,c\in R\)

 8.  \(3\left( {{a}^{2}}+{{b}^{2}}+\left. {{c}^{2}} \right) \right.\ge {{\left( a+b+c \right)}^{2}}\) \(\forall a,b,c\in \)\(R\)

 9.  \(\frac{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}{a+b+c}\ge \frac{1}{3}\left( a+b+c \right)\) \(\forall \)\(a,b,c\in R\)

 10.  \(\sqrt{a+b+c}\ge \frac{\sqrt{3}}{3}\left( \sqrt{a}+\sqrt{b}+\sqrt{c} \right)\forall a,b,c\ge 0\)

 11.  \(\left( n-1 \right)\left( a_{1}^{2}+...+a_{n}^{2} \right)\ge 2\left( {{a}_{1}}{{a}_{2}}+...{{a}_{1}}{{a}_{n}}+{{a}_{2}}{{a}_{3}}+...+{{a}_{n-1}}{{a}_{n}} \right)\)\)\(

 13.  \(\frac{{{a}^{n}}+{{b}^{n}}}{2}\ge {{\left( \frac{a+b}{2} \right)}^{2}},\forall n\in N,a,b>0.\)

 14.  \(0<\frac{a}{b}<2\Rightarrow \frac{a}{b}<\frac{a+r}{b+r},\forall r>0.\)

        \(1<\frac{a}{b}\Rightarrow \frac{a}{b}>\frac{a+r}{b+r},\forall r>0\)

 15.  \(\left| x \right|\)\(\le a\) \(\left( a>0 \right)\) \(\Leftrightarrow \)\(-a\le x\le a.\)

 16.  \(\left| a\pm b \right|\le \left| a \right|+\left| b \right|\[,\]a,b\in R\) \(sauC\).

 17.  \(\left| {{a}_{1}}\pm {{a}_{2}}\pm ...\pm {{a}_{n}} \right|\le \left| {{a}_{1}} \right|+...+\left| {{a}_{n}} \right|\) \(,\) \(in\[R\]sau\[C\].\)

 18.  \(\left| \left| a \right|-\left| b \right| \right|\le \left| a-b \right|\[in\]R\[sau\]C\)\(.\)

 19.  \(\frac{1}{{{n}^{2}}}=\frac{1}{n\cdot n}\le \frac{1}{\left( n-1 \right)n}=\frac{1}{n-1}-\frac{1}{n}\)

        \(\frac{1}{n!}<\frac{1}{\left( n-1 \right)n}=\frac{1}{n-1}-\frac{1}{n}\)              

 

20. \(a,b\in Z\) ,\(m,n\in Z\) \(,\[\sqrt[{}]{\frac{m}{n}}\notin Q\]\Rightarrow \left| m{{a}^{2}}-n{{b}^{2}} \right|\ge 1.\)

21. Numerele pozitive \(a,b,c\)pot fi lungimile laturilor unui triunghi  dacă şi numai dacă \(\exists \) \(x,y,z\in R_{+}^{*}\)\(a.i\)   \(a=y+z,\[b=x+z,\]c=x+y.\)

22.  \({{\left( \frac{a}{b} \right)}^{a-b}}\ge 1\) \(a\ne b\) \(\forall \)\(a,b>0\),

23. \(a,b,c\in R_{+}^{*}\Rightarrow \frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}\ge 6.\)

24. Dacă\({{x}_{1}},...,{{x}_{n}}\ge 0\)si \({{x}_{1}}+...+{{x}_{n}}=k\)constant atunci produsul \({{x}_{2}}\cdot {{x}_{2}}...{{x}_{n}}\)e maxim când \({{x}_{1}}=...={{x}_{n}}=\frac{k}{n}.\)\)\(