If $a, b, c, d$ are positive real numbers such that $a+b+c+d=2$, then $M=(a+b)(c+d)$ satisfies the r

Question

If $a, b, c, d$ are positive real numbers such that $a+b+c+d=2$, then $M=(a+b)(c+d)$ satisfies the relation\

Professor PC Thomas Classes · Answer

(a) : As A.M. $\geq$ G.M. for positive real numbers, we get\ $\frac{(a+b)+(c+d)}{2} \geq[(a+b)(c+d)]^{1 / 2}$\ $\Rightarrow \quad M \leq 1 \quad(\because a, b, c, d>0)$\ Also $(a+b)(c+d)>0 \Rightarrow 0<M \leq 1$.