Question

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

• A.
• B.
• C.
• D.

Answer 1

Answer: (A), (C), (D)

The correct options are
A

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

Since, $\frac{a+b}{4}\le \frac{a+b}{2}$, $⟹\frac{ab}{a+b}\le \frac{a+b}{2}$.

So, option A is also correct.

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