If x and y are positive real numbers and xy = 8, then the minimum value of 2x + y is?

Solution(By Examveda Team)

$$\eqalign{ & xy = 8{\text{ }}\left( {{\text{Given}}} \right) \cr & {\text{So, }}\left( {x,y} \right) = \left( {1,8} \right) \cr} $$
We have to question the options and check them.
$$\eqalign{ & \left( {8,1} \right) \cr & \left( {2,4} \right) \cr & \left( {4,2} \right) \cr & \therefore {\text{ }}2x + y \cr & = 2 \times 1 + 8 \cr & = 10 \cr & 2 \times 8 + 1 = 11 \cr & 2 \times 2 + 4 = 8{\text{ }}\left( {{\text{Minimum}}} \right) \cr & 2 \times 4 + 2 = 10 \cr} $$
Hence, in this question we have all the options.
So, take all the positive factor otherwise we should have to take - ve(negative) values also.
$$\eqalign{ & \left( {x,y} \right) = \left( {1,8} \right) \cr & {\text{ }}\left( {8,1} \right) \cr & {\text{ }}\left( {2,4} \right) \cr & {\text{ }}\left( {4,2} \right) \cr & {\text{ }}\left( { - 1, - 8} \right) \cr & {\text{ }}\left( { - 8, - 1} \right) \cr & {\text{ }}\left( { - 2, - 4} \right) \cr & {\text{ }}\left( { - 4, - 2} \right) \cr} $$