About Lesson
14.3 Proof of Inequalities 不等式的证明
Prove the following inequalities 证明以下不等式
(1)
>
, ![Rendered by QuickLaTeX.com x \neq y \neq z](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-996ed97614d5fd7aee7a58a8671448f4_l3.png)
(1)
![Rendered by QuickLaTeX.com y^2z^2 + z^2x^2 + x^2y^2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-ae696636ec8c57c57b72cfd86cd75070_l3.png)
![Rendered by QuickLaTeX.com xyz (x + y + z)](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-bafa7e2e515143c3efaaba08615e916b_l3.png)
![Rendered by QuickLaTeX.com x \neq y \neq z](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-996ed97614d5fd7aee7a58a8671448f4_l3.png)
(2)
, x, y > 0
![Rendered by QuickLaTeX.com \frac{x}{y^2}+\frac{y}{x^2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-906c9ba8d396ad7ae0a8a94d1492f323_l3.png)
![Rendered by QuickLaTeX.com \geq](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-0bea0a8ab6cbdbdabafec0fa25e57e60_l3.png)
![Rendered by QuickLaTeX.com \frac{1}{x}+\frac{1}{y}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-60bbd30492dc27959eaafb2f5f8ddfed_l3.png)
(3) >
,
(4) Prove that
,
Hence or otherwise, prove that (x + y + z)2
3 (xy + yz + xz)
![Rendered by QuickLaTeX.com x^2 + y^2 + z^2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-1fdfcea45f1c47a2cfe445a9a4211a22_l3.png)
![Rendered by QuickLaTeX.com \geq](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-0bea0a8ab6cbdbdabafec0fa25e57e60_l3.png)
![Rendered by QuickLaTeX.com xy + yz + xz](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-f044d4fa0f9144f7e500e6283aef33bc_l3.png)
Hence or otherwise, prove that (x + y + z)2
![Rendered by QuickLaTeX.com \geq](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-0bea0a8ab6cbdbdabafec0fa25e57e60_l3.png)
证明
,
据此或其他方法,证明 (x + y + z)2 3 (xy + yz + xz)