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Question Number 109595 by nimnim last updated on 24/Aug/20

$$ForanyRealnumbersx,yandz,$$$$if(x+y+z)=2,thenprovethat$$$$xyz⩾8(1-x)(1-y)(1-z)$$

Commented by 1549442205PVT last updated on 25/Aug/20

$$\mathrm{This}\mathrm{inequality}\mathrm{is}\mathrm{only}\mathrm{true}\mathrm{for}$$$$\mathrm{x},\mathrm{y},\mathrm{z}>0.\mathrm{Example},\mathrm{for}\mathrm{x}=-\frac{2}{3},\mathrm{y}=\frac{6}{5}$$$$\mathrm{z}=\frac{22}{15}\mathrm{then}\mathrm{it}\mathrm{is}\mathrm{false}.\mathrm{Then}$$$$\mathbf{xyz}-8(1-\mathbf{x})(1-\mathbf{y})(1-\mathbf{z})=-\frac{544}{25}<0$$

Answered by floor(10²Eta[1]) last updated on 24/Aug/20

$$\frac{\mathrm{x}+\mathrm{y}+\mathrm{z}}{3}{⩾}^3\sqrt{\mathrm{xyz}}\Rightarrow \frac{2}{3}{⩾}^3\sqrt{\mathrm{xyz}}\therefore \frac{8}{27}⩾\mathrm{xyz}$$$$\Rightarrow \mathrm{xyz}⩾8(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})$$$$\iff \frac{1}{27}⩾(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})$$$$\frac{(1-\mathrm{x})+(1-\mathrm{y})+(1-\mathrm{z})}{3}{⩾}^3\sqrt{(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})}$$$$\frac{3-(\mathrm{x}+\mathrm{y}+\mathrm{z})}{3}{⩾}^3\sqrt{(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})}$$$$\frac{1}{3}{⩾}^3\sqrt{(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})}$$$$\frac{1}{27}⩾(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z})$$$$\therefore \mathrm{xyz}⩾8(1-\mathrm{x})(1-\mathrm{y})(1-\mathrm{z}),\mathrm{x}+\mathrm{y}+\mathrm{z}=2$$

Commented by nimnim last updated on 24/Aug/20

$$Thankyousir..$$

Commented by floor(10²Eta[1]) last updated on 25/Aug/20

$$\mathrm{i}\mathrm{just}\mathrm{used}\mathrm{AM}-\mathrm{GM}\mathrm{inequality}\mathrm{which}$$$$\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{all}\mathrm{real}\mathrm{numbers}$$

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