x-y-z-R-a-b-c-gt-0-prove-that-x-2-a-y-2-b-z-2-c-x-y-z-2-a-b-c-

Question Number 201615 by hardmath last updated on 09/Dec/23

x, y, z \in R

a, b, c > 0

prove that :

\frac{x^{2}}{a} + \frac{y^{2}}{b} + \frac{z^{2}}{c} ⩾ \frac{{(x + y + z)}^{2}}{a + b + c}

Answered by AST last updated on 09/Dec/23

\sqrt{{(\frac{x}{\sqrt{a}})}^{2} + {(\frac{y}{\sqrt{b}})}^{2} + {(\frac{z}{\sqrt{c}})}^{2}} \sqrt{{(\sqrt{a})}^{2} + {(\sqrt{b})}^{2} + {(\sqrt{c})}^{2}}

⩾ (x + y + z)

∣ u ∣∣ v ∣⩾ u \cdot v w h e r e u = ⟨ \frac{x}{\sqrt{a}}, \frac{y}{\sqrt{b}}, \frac{z}{\sqrt{c}} ⟩; v = ⟨ \sqrt{a}, \sqrt{b}, \sqrt{c} ⟩

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