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Albert1
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a,b,c,d > 0 , please prove :
$ \sqrt{a+b+c+d} \geq \dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}{2}$
$ \sqrt{a+b+c+d} \geq \dfrac{\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}}{2}$
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this is a very elagant and simple proof.Albert said:By Cauchy-Schwarz inequality:
$({1}^{2}+{1}^{2}+{1}^{2}+{1}^{2})({\sqrt{a}}^{2}+{\sqrt{b}}^{2}+{\sqrt{c}}^{2}+{\sqrt{b}}^{2})
\geq (\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d})^2$$4(a+b+c+d)\geq (\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d})^2 $
$ \therefore \sqrt{a+b+c+d}\geq \dfrac{(\sqrt{a}+\sqrt{b}++\sqrt{c}++\sqrt{d})}{2}$
Proving an inequality means demonstrating that one expression is either greater than, less than, or equal to another expression. This is typically done using mathematical techniques and logical reasoning.
Proving inequalities is important because it allows us to establish the relationships between different mathematical expressions. This can help us to solve problems and make accurate conclusions in various fields such as economics, physics, and engineering.
There are several techniques for proving inequalities, including algebraic manipulation, using known inequalities, and using mathematical theorems and properties. It is important to carefully follow the rules of mathematical reasoning and provide clear and logical steps in the proof.
Yes, some common mistakes to avoid when proving inequalities include incorrect use of algebraic rules, skipping steps, and using circular reasoning. It is also important to be careful with inequalities involving variables, as they may only be valid for certain values of those variables.
Yes, inequalities can be proved using real-life examples, but they must also be supported by mathematical reasoning. Real-life examples can help to illustrate the practical applications of inequalities and make them more understandable, but they cannot be used as the sole basis for a proof.