If f ( x ) = ( tan 1 ) x + log e ( 123 ) log e ( 1234 ) - ( tan 1 ) , x 0 , then the least value of f ( f ( x ) ) + f ( f ( 4 x ) ) is [2023]

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Solution

Q.
If $f (x) = \frac{(\tan 1^{o}) x + \log_{e} (123)}{x \log_{e} (1234) - (\tan 1^{o})}, x > 0,$ then the least value of $f (f (x)) + f (f (\frac{4}{x}))$ is [2023]

1 4

2 2

3 0

4 8

Ans.
(1)

Let $f (x) = \frac{A x + B}{C x - A}$

$f (f (x)) = \frac{A (\frac{A x + B}{C x - A}) + B}{C (\frac{A x + B}{C x - A}) - A} = \frac{A^{2} x + B C x}{B C + A^{2}} = x \dots (i)$

$f (f (\frac{4}{x})) = \frac{4}{x} [From (i)]$

$∴ f (f (x)) + f (f (\frac{4}{x})) = x + \frac{4}{x} \geq 4 [Using A.M. \geq G.M.]$

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