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

$$\mathrm{x},\mathrm{y},\mathrm{z}\in \mathbb{R}$$$$\mathrm{a},\mathrm{b},\mathrm{c}>0$$$$\mathrm{prove}\mathrm{that}:$$$$\frac{{\mathrm{x}}^2}{\mathrm{a}}+\frac{{\mathrm{y}}^2}{\mathrm{b}}+\frac{{\mathrm{z}}^2}{\mathrm{c}}⩾\frac{{(\mathrm{x}+\mathrm{y}+\mathrm{z})}^2}{\mathrm{a}+\mathrm{b}+\mathrm{c}}$$

Answered by deleteduser1 last updated on 09/Dec/23

$$\sqrt{{\left(\frac{x}{\sqrt{a}}\right)}^2+{\left(\frac{y}{\sqrt{b}}\right)}^2+{\left(\frac{z}{\sqrt{c}}\right)}^2}\sqrt{{\left(\sqrt{a}\right)}^2+{\left(\sqrt{b}\right)}^2+{\left(\sqrt{c}\right)}^2}$$$$⩾(x+y+z)$$$$\mid u\mid \mid v\mid ⩾u\cdot vwhereu=⟨\frac{x}{\sqrt{a}},\frac{y}{\sqrt{b}},\frac{z}{\sqrt{c}}⟩;v=⟨\sqrt{a},\sqrt{b},\sqrt{c}⟩$$

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