Friday, May 22, 2015

Compare M and N

Given that $p,\,q,\,r,\,s,\,a,\,b$ are positive real numbers with $M=\sqrt{ap+br}\cdot \sqrt{\dfrac{q}{a}+\dfrac{s}{b}}$ and $N=\sqrt{pq}+\sqrt{rs}$. Compare $M$ and $N$.

This question has the best form of quality because it allows us to approach it in at least 3 different ways, 3 of which that are enlighting and inspiring.

My solution:

Applying the Cauchy Schwarz inequality with $\sqrt{ap+br}$ and $\sqrt{\dfrac{q}{a}+\dfrac{s}{b}}$, i.e., the two factors of M, we get:

$\begin{align*}M&=\sqrt{ap+br}\cdot \sqrt{\dfrac{q}{a}+\dfrac{s}{b}}\\&\ge \sqrt{ap}\sqrt{\dfrac{q}{a}}+\sqrt{br}\sqrt{\dfrac{s}{b}}\\&\ge \sqrt{\cancel{a}p}\sqrt{\dfrac{q}{\cancel{a}}}+\sqrt{\cancel{b}r}\sqrt{\dfrac{s}{\cancel{b}}}\\&\ge \sqrt{pq}+\sqrt{rs}\\&\ge N\end{align*}$

Therefore, $M\ge Q$.

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