Question
If $$a, b, c, d$$ are positive real numbers such that $$a + b + c + d = 2,\,{\text{then }}M = \left( {a + b} \right)\left( {c + d} \right)$$ satisfies the relation
A.
$$0 \leqslant M \leqslant 1$$
B.
$$1 \leqslant M \leqslant 2$$
C.
$$2 \leqslant M \leqslant 3$$
D.
$$3 \leqslant M \leqslant 4$$
Answer :
$$0 \leqslant M \leqslant 1$$
Solution :
As A.M. $$ \geqslant $$ G,M, for positive real numbers, we get
$$\eqalign{
& \frac{{\left( {a + b} \right) + \left( {c + d} \right)}}{2} \geqslant \sqrt {\left( {a + b} \right)\left( {c + d} \right)} \cr
& \Rightarrow \,M \leqslant 1 \left( {{\text{Putting values}}} \right) \cr
& {\text{Also }}\left( {a + b} \right)\left( {c + d} \right) > 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\therefore \,\,a,b,c,d > 0} \right] \cr
& \therefore \,\,0 \leqslant M \leqslant 1 \cr} $$