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If $a,b$ are positive real variables whose sum is a constant $\lambda$, then the minimum value of $\sqrt{(1+1/a)(1+1/b)}$ is

1. $\lambda \: – 1/\lambda$
2. $\lambda + 2/\lambda$
3. $\lambda+1/\lambda$
4. None of the above

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Can you please check, there must be a brackets in $B$ option.

@Arjun Sir

Like $(\frac{\lambda + 2}{\lambda})$

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Are the first three options like below?
$\mathrm{A.}~\frac{\lambda-1}{\lambda}$
$\mathrm{B.}~\frac{\lambda+2}{\lambda}$
$\mathrm{C.}~\frac{\lambda+1}{\lambda}$
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Yeah it should be like that only.

Mistake may have occured while editing.
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I think so.
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Answer: $\mathbf B$

Explanation:

Let $a + b = \lambda$

$\therefore b = \lambda - a$

Let $f = \sqrt{({1 + \frac{1}{a}})(1 + \frac{1}{b})}$

Let $g = f^2 =({1 + \frac{1}{a}})(1 + \frac{1}{b})$

Taking derivative of $g$ with respect to $a$ and making it equal to $0$, we get:

$g' = \frac{-1}{a^2}(1+\frac{1}{\lambda-a}) + (1+\frac{1}{a})(\frac{1}{(\lambda-a)^2}) = 0$

Simplifying this we get:

$(\lambda-2a)(\lambda+1) = 0, \;or \; a = \frac{\lambda}{2}$

Again on differentiating $g$ with respect to a, we get:

$g^{''} = (\frac{2}{a^3})(1+\frac{1}{\lambda-a}) - \frac{1}{a^2}\frac{1}{(\lambda-a)^2} + (1+\frac{1}{a})(\frac{2}{(\lambda-a)^3})-\frac{1}{a^2}(\frac{1}{(\lambda-a)^2})$

At $a = \frac{\lambda}{2}$ the value of $g^{''}>0$

$\Rightarrow$ It's a point of minima.

$\therefore$ Minimum value $= \frac{\lambda+2}{\lambda}$

$\therefore \mathbf B$ is the correct option.

by Boss (13.4k points)
edited by
+1

This line below has some mistakes. Perhaps typing ones. Please make corrections. 0
What's that?
+1
Yes ok.

got it.

thanks.

Edit: Corrected.