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Question Number 88364 by Power last updated on 10/Apr/20

	Answered by mind is power last updated on 10/Apr/20
	$by Cauchy shwart Σ_(i=1) ^n 1.x_i ≤(√Σ_(i=1) ^n )1^2 .(√(Σ_(i=1) ^n x_i ^2 )),.....1 Σx_i =1 Σx_i ^2 =(1/n) 1⇔1≤(√n).(1/(√n))=1 equality⇒by cauchy shwartz (x_1 ,x_2 ,....,x_n )=(a,a,....,a) ⇒na=1⇒a=(1/n),∀i∈{1,....n} x_i =(1/n)$
	$b y C a u c h y s h w a r t$ $\underset{i = 1}{\sum^{n}} 1 . x_{i} ⩽ \sqrt{\underset{i = 1}{\sum^{n}}} 1^{2} . \sqrt{\underset{i = 1}{\sum^{n}} x_{i}^{2}}, . . . . . 1$ $Σ x_{i} = 1$ $Σ x_{i}^{2} = \frac{1}{n}$ $1 \Leftrightarrow 1 ⩽ \sqrt{n} . \frac{1}{\sqrt{n}} = 1 e q u a l i t y \Rightarrow b y c a u c h y s h w a r t z$ $(x_{1}, x_{2}, . . . ., x_{n}) = (a, a, . . . ., a)$ $\Rightarrow n a = 1 \Rightarrow a = \frac{1}{n}, \forall i \in {1, . . . . n} x_{i} = \frac{1}{n}$
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