Question

If x, y, z $ϵ$ $R^{+}$ [ $R^{+}$ denote real positive numbers] Then, $\frac{ab}{a+b}$ is

• A.
• B.
• C.
• D.

Answer 1

Answer: (C), (D)

The correct options are
C

D

Observe that $\frac{ab}{a+b}$ is a part of HM of a & b i.e., HM = $\frac{2ab}{a+b}$

Other relations in options are recognized as AM & GM.

As we know AM $\ge$ GM $\ge$ HM

(i) HM $\le$ GM

$\Rightarrow$ $\frac{2ab}{a+b}$ $\le$ ${\left(ab\right)}^{\frac{1}{2}}$

$\Rightarrow$ $\frac{ab}{a+b}$ $\le$ $\frac{{\left(ab\right)}^{\frac{1}{2}}}{2}$----------(1)

So, option C is true

(iii) HM $\le$ AM

$\frac{2ab}{a+b}$ $\le$ $\frac{a+b}{2}$

$\frac{ab}{a+b}$ $\le$ $\frac{a+b}{4}$ --------------------------------(2)

So, option D is also true

From (1) & (2), right options are C, D