<s>[INST] Show that $(b_1+b_2)^2\leq (a_1+a_2)(c_1+c_2)$ How to show that if $a_k,b_k,c_k\in \mathbb{R}^+$ for which $a_kc_k-b_k^2\geq 0$ for $k=1,2$, then
$$
(b_1+b_2)^2\leq (a_1+a_2)(c_1+c_2)?
$$
I tried to check some well-known identities such as Young's inequality, but I failed to make it work.
 [/INST] HINT
You can start with the RHS as follows
\begin{align*}
(a_{1} + a_{2})(c_{1} + c_{2}) & = a_{1}c_{1} + a_{1}c_{2} + a_{2}c_{1} + a_{2}c_{2}\\\\
& \geq b^{2}_{1} + a_{1}c_{2} + a_{2}c_{2} + b^{2}_{2}
\end{align*}
Now you can apply the AM-GM inequality in order to obtain
\begin{align*}
a_{1}c_{2} + a_{2}c_{1} & \geq 2\sqrt{a_{1}c_{2}a_{2}c_{1}}\\\\
& = 2\sqrt{a_{1}c_{1}a_{2}c_{2}}\\\\
& \geq 2\sqrt{b^{2}_{1}b^{2}_{2}}\\\\
& = 2b_{1}b_{2}
\end{align*}
Can you take it from here?
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