# More AM-GM

Prove that if $$x+y=1$$ then $$(1+\frac{1}{x})(1+\frac{1}{y})\geq9$$.

Note by Joshua Ong
4 years, 2 months ago

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Cauchy Schwarz gives $\left(1+\dfrac{1}{x}\right)\left(1+\dfrac{1}{y}\right)\ge \left(1+\dfrac{1}{\sqrt{xy}}\right)^2$

However, $$\dfrac{1}{\sqrt{xy}}\ge \dfrac{2}{x+y}=2$$ by AM-GM, so $\left(1+\dfrac{1}{x}\right)\left(1+\dfrac{1}{y}\right)\ge \left(1+\dfrac{1}{\sqrt{xy}}\right)^2\ge (1+2)^2=9$ and we are done.

- 4 years, 2 months ago